Spectral and scattering theory of spatially cut-off P (φ)_{2} Hamiltonians

J. Dereziński Department of Mathematical Methods in Physics, Warsaw University, Hoża 74, 00-682 Warszawa, Poland.

C. Gérard Centre de Mathématiques, URA 169 CNRS, Ecole Polytechnique 91128 Palaiseau Cedex, France

November 1998

Abstract We study spatially cut-off P (φ)_{2} Hamiltonians. We show the local finiteness of the pure point spectrum outside of thresholds, the limiting absorption principle and asymptotic completeness of scattering for such Hamiltonians. Our results imply the absence of singular continuous spectrum.

1 Introduction

1.1 P (φ)_{2} models in quantum field theory

Models of quantum field theory used by physicists to describe basic interactions, although very successful experimentally, are defined only in a formal and perturbative way. In year 1952 Wightman and Gårding formulated a set of axioms, which, at least at that time, seemed to constitute a rather general mathematical framework for a physically acceptable QFT of basic interactions. In particular, these axioms satisfied the requirements of relativistic covariance and causal locality. It was hoped that physically realistic models of QFT can be interpreted in a mathematically consistent way and that they can be shown to satisfy axioms similar to Wightman axioms. At that time no examples of theories satisfying Wightman axioms were known except for free fields, which are in a sense trivial both from the physical and mathematical point of view.

It is not difficult to give a list of non-trivial QFT models which on a formal level satisfy Wightman axioms. These models can be ordered according to their difficulty and physically realistic models in 3+1 dimensions are quite high on this list. Wightman proposed to construct these models one by one and check whether they satisfy the axioms he formulated, starting with the easiest (but unfortunately, non-physical) ones. Thus began one of the most famous chapters of mathematical physics – constructive quantum field theory.

The simplest class of models in the Wightman program were the so-called P (φ)_{2} models, that is the models of self-interacting bosons in 2 space-time dimensions with the interaction given by a semibounded polynomial P (φ) of degree at least 4. The construction of these models was one of the early successes of constructive field theory. A number of different constructions were given. One of the approaches (in fact, the one that was used in the earliest works) started with considering a spatially cutoff P (φ)_{2} interaction, where the cutoff is defined using a positive coupling function g(x), which decays sufficiently fast at infinity. One can then define the Hamiltonian H := H_{0} + ∫ g(x) :P (φ(x)): dx, (1.1)

as a semibounded self-adjoint operator on Fock space Γ(L^{2}(IR)), where H_{0} = dΓ(\sqrt{k^{2} + m^{2}}) is the free Hamiltonian. (1.1) is called the spatially cut-off P (φ) Hamiltonian. The next step is to show that, as g(x) → 1, one obtains a limiting dynamics which acts in a different, renormalized Hilbert space and satisfies the Wightman axioms.

The Hamiltonian H will be the main subject of our paper. We will always assume that g ∈ L^{1}(IR); for most results we will also need some additional assumptions on the decay and differentiability of g.

The program of constructive field theory has not attained its original goal of constructing a physically realistic and mathematically rigorous model satisfying the conditions of covariance and locality in 4 space-time dimensions. To our knowledge, the models that have been constructed, including P (φ)_{2} do not describe any real physical systems. Nevertheless, we believe that the heritage of constructive field theory is a source of models and techniques which are very interesting both physically and mathematically.

One could ask what are the reasons to look at the Hamiltonians (1.1). One of them is historic – as we tried to sketch above, these Hamiltonians played an important role in the development of constructive field theory and there is a considerable literature on this subject. Unfortunately, (1.1) is not relativistic, since it is not even translation invariant. Nevertheless, it has a certain remarkable property: it satisfies the axiom of the causal locality, more precisely, if one defines a net of local algebras in the sense of Haag–Kastler with help of H, then this net is causally local.

Another reason is that spatially cutoff P (φ)_{2} Hamiltonians can be viewed as examples of Schrödinger operators in infinite dimension. Studying such Hamiltonians is a good occasion to test various advanced tools of functional analysis and sheds light on the mathematical structure of quantum field theory. This point of view was advocated in Simon’s survey [Si2], where a number of mathematical questions concerning the spectral theory of P (φ)_{2} Hamiltonians are formulated.

1.2 Content of this paper

In the present paper we extend methods developed for N-body Schrödinger operators to study spatially cut-off P (φ)_{2} Hamiltonians. Our results include

1) local finiteness of the pure point spectrum outside of the thresholds,

2) the limiting absorption principle,

3) asymptotic completeness. Note that 2) and 3) imply the absence of the singular continuous spectrum. (The properties 2) and 3) are proven under different assumptions, neither of which implies the other).

Recently a number of papers appeared that study other models that belong to a broadly understood QFT [AH, BFS, BFSS, DG1, Ge, HuSp1, HuSp2, JP1, JP2, Sk, Sp1, Sp2]. The Hamiltonians studied in these papers are sometimes called Pauli-Fierz Hamiltonians. They are non-relativistic, non-local and they have little to do with the Wightman program. Nevertheless, they are physically relevant and of a significant mathematical interest.

One of these papers – [DG1] – can be viewed as a predecessor of this paper. It is devoted to massive Pauli-Fierz Hamiltonians and it contains results similar to those contained in this paper (except for the limiting absorption principle, which, however, could be easily shown in the context of [DG1]). It should be noted that there are a lot of analogies between this paper and [DG1]. Both massive Pauli-Fierz Hamiltonians and spatially cut-off P (φ)_{2} models share a lot of common characteristics, in particular the basic framework of scattering theory is essentially the same. Both classes are examples of QFT Hamiltonians with localized interactions.

Nevertheless, the technical difficulties of this paper are more serious than those of [DG1]. This is partly due to the fact that it is much more difficult to define a P (φ)_{2} Hamiltonian than a Pauli-Fierz Hamiltonian. In the case of Pauli-Fierz Hamiltonians considered in [DG1], the perturbation is relatively bounded, which is not true in the case of (1.1). These problems become especially apparent when one considers the Mourre estimate, which requires a rather careful treatment and in many respects is more difficult than in the case of Schrödinger operators. In fact, the original theory of Mourre [Mo] does not seem to be applicable in the case of H and we need to apply its more sophisticated version contained in [ABG]. The key idea of the approach of [ABG] is the property of C^{1}(A) regularity of an Hamiltonian H with respect to a unitary group e^{isA}, which fortunately can be verified in the case of P (φ)_{2} Hamiltonians.

The main tools that we use in the study of H is considering the interaction as an operator of multiplication in the Q-representation and the higher order estimates due to Rosen. These tools were developed in the early years of constructive field theory [Ne, Ro1, Ro2, Se, S-H.K]. Note that these tools are not needed in the case of technically simpler Pauli-Fierz Hamiltonians considered in [DG1].

Another difference between this paper and [DG1] is a significant simplification of the proof of asymptotic completeness and a different proof of the Fock property of asymptotic fields.

Our paper can be divided into two parts. The first part, which consists of Sects. 2, 3, 4, 5 describes the general formalism of CAR representations, bosonic Fock spaces and Q−space representation. Our presentation is quite general and at some points its generality goes beyond what we need in the case of the Hamiltonians (1.1). Actually, when one considers some other models of QFT (such as those with massless particles) one needs the formalism in its more general form (see for instance Theorem 4.3, which allows for a non-Fock component of CCR representations). Most of the material of these sections can be found in the literature, notably Sects. 2 and 4 follow quite closely [BR] and Sect. 5 follows [S-H.K, Si1]. Nevertheless, our presentation has some modifications and improvements as compared for example to that in [BR] and we believe that the reader will find it useful, especially since it is compact and essentially self-contained.

A considerable effort has been devoted to develop a concise notation for operators in Fock spaces. Some elements of this notation are standard (due in particular to I. Segal), others were introduced in [DG1]. In [DG1] we did not need to consider Wick polynomials, which play an important role in this paper. We devote a special attention to the properties of Wick polynomials in Subsect. 3.12, and also in the context of the Q-representation in Subsection 5.2. Note that the calculus and notation in the literature on QFT can be quite cumbersome and ad hoc, which we wanted to avoid.

The second part of our paper is devoted to the study of spatially cutoff P (φ)_{2} Hamiltonians. In Section 6 we introduce spatially cut-off P (φ)_{2} Hamiltonians and we describe their basic properties, following eg. [S-H.K].

One of the most difficult results about such Hamiltonians are the so-called higher order estimates due to Rosen. They are described with some of their consequences in Sect. 7. Strictly speaking, their proof contained in [Ro2] does not cover the class of Hamiltonians that we consider. Therefore, we indicate how to modify the arguments of [Ro2] to cover our class of coupling functions g.

In Section 8 we study the commutator of H with the second quantized generator of dilations A. The operator A will play the role of a conjugate operator in the Mourre theory. The abstract framework of this section is based on [ABG], where a theory of the C^{1}(A) property is developed. Such a careful treatment of this question was not needed neither in [DG1] nor in the case of N−particle Schrödinger operators.

The case of the φ_{2}^{4} model, ie the case when P is a polynomial of degree 4 is simpler. For example the construction of the space cutoff φ_{2}^{4} model can be done without using the Q−representation (see [GJ2]). Similarly the Mourre theory in the φ_{2}^{4} case can be treated in a simpler way, under weaker conditions on the cutoff function g.

Sect. 9 is devoted to the spectral theory of P (φ)_{2} Hamiltonians. The analog of the HVZ theorem is proven in Subsect. 9.1. This result was first proven in [GJ4, S-H.K]; we give a different proof (analogous to the one given in [DG1]), which is essentially a by-product of the techniques which we develop in our paper for other purposes.

In Subsection 9.2 we prove the Mourre estimate for H. The proof is similar to the one contained in [DG1]. The Mourre estimate implies the local finiteness of the pure point spectrum outside of the thresholds. The set of thresholds is defined as {λ+nm | n = 1.2, …, λ ∈ σ_{pp}(H)}, where σ_{pp}(H) denotes the pure point spectrum of H. Note that this result implies that the pure point spectrum of H is contained in a closed set of measure zero.

Under stronger conditions on the coupling function, we can also show the limiting absorption principle, which implies the absence of singular continuous spectrum. More precisely, we show the existence of the boundary value of the resolvent on the real line

lim_{ϵ↓0}⟨A⟩^{−μ}(λ + iϵ − H)^{−1}⟨A⟩^{−μ},

where A is the conjugate operator and μ > \frac{1}{2}.

In Section 10 we study the scattering theory for spatially cut-off P (φ)_{2} Hamiltonians. The basic construction of scattering theory in the context of this paper are asymptotic fields, that is the limits of the field operators in the interaction picture:

a^{±,#}(g) := s- lim_{t→±∞} e^{itH}a^{#}(g_{t})e^{−itH},

where g_{t} := e^{−it\sqrt{k^{2}+m^{2}}}g. We prove the existence of the asymptotic fields and show that they realize a CCR representation satisfying the Fock property. This result was first proven in [HK]. Up to technical details due to a more singular character of the interaction, the proof of the existence of asymptotic fields follows the proof of the analogous result in [DG1]. The proof of the Fock property is based on the general theory of CCR representations. Its main ingredient is the concept of the number operator associated to regular CCR representation described in Section 4. Note that the proof of the Fock property contained in [DG1] was different – it was closer to the original argument of [HK].

With the CCR representations given by a^{±,#}(g) one can associate the spaces of asymptotic vacua K^{±}, that is the states annihilated by asymptotic annihilation operators. The Fock property is equivalent to saying that vectors of the form a^{±∗}(g_{1}) ⋯ a^{±∗}(g_{n})ψ, where ψ ∈ K^{±}, span the whole Hilbert space H. It is easy to see that the bound states are contained in the spaces of the asymptotic vacua K^{±}. The property of asymptotic completeness means that the spaces K^{±} are equal to the space of bound states of H. This property is formulated at the end of Sect.

10. Among its consequences are the fact that the asymptotic vacua at t = −∞ and at t = +∞ coincide and the justification of the formalism of asymptotic states commonly used by physicists.

In Section 11 we describe various propagation estimates. Their proofs do not differ substantially from the Pauli-Fierz case and we refer to [DG1] for most of them.

In Sect. 12 we prove asymptotic completeness, that is, we show that the space of asymptotic vacua equals the pure point subspace of H. In principle, in this section we could repeat almost verbatim the arguments of the analogous section of [DG1]. Nevertheless, we simplify substantially the arguments of [DG1]. The major difference is that in [DG1] we used operators P_{k}, Q_{k} and their asymptotic counterparts. In this paper we avoid using them and the main role is played by the operators Γ^{+}(q) (describing something similar to the asymptotic velocity) and inverse wave operators W^{+}(j) (both of these objects were also used in [DG1]). Note in parenthesis that the absence of the operators P_{k} in the present paper has its price – it seems that one needs them to show a certain interesting intermediate result concerning the inverse wave operators W^{+}(j) (see [DG1, Thm 7.13]), which, fortunately, is not needed for the proof of asymptotic completeness itself.

The methods of this paper can be applied to other models of QFT with a localized interaction and a massive dispersion relation. In particular, one can use ideas of this paper to simplify some of the arguments of [DG1].

As we mentioned earlier, the P (φ)_{2} models are the simplest nontrivial models considered in constructive field theory. Still, their treatment requires a lot of care and involves a number of various techniques, which go beyond the problems usually encountered in quantum mechanics and PDE’s. Even more difficult and more interesting problems arise when one considers other models of constructive field theory such as Y_{2} or λφ_{3}^{4}. It would be interesting to extend our results to spatially cut-off versions of these models. We believe that it is feasible, since they are also models with a localized interaction and a massive dispersion relation. In particular, the framework of scattering theory for these models is essentially the same as the one considered in this paper. The main new difficulty would be the various renormalization procedures needed to define these Hamiltonians. Acknowledgements The research of the first author was a part of the project Nr 2 P03A 019 15 financed by a grant of Komitet Badań Naukowych. Part of this work was done during a visit of the first author at the Aarhus University supported by MaPhySto funded by the Danish National Research Foundation.

2 CCR Representations

We recall some standard facts on CCR representations.

2.1 Weyl operators

Let H be a Hilbert space. Let 𝔤 be a real vector space equipped with an antisymmetric form σ. A map 𝔤 ∋ h ↦ W_{π}(h) ∈ B(H) (2.1)

is a representation of the canonical commutation relations (in short a CCR representation) over 𝔤 in H if

\begin{matrix}W_{π}(h_{1})W_{π}(h_{2}) = e^{−iσ(h_{1},h_{2})/2}W_{π}(h_{1} + h_{2}), \\ W_{π}^{∗}(h) = W_{π}(−h), W_{π}(0) = 1l. \end{matrix} (2.2)

Note that as a consequence W_{π}(h) are unitary and we have

W_{π}(h_{1})W_{π}(h_{2}) = e^{−iσ(h_{1},h_{2})}W_{π}(h_{2})W_{π}(h_{1}). (2.3)

2.2 Field operators

We say that the CCR representation (2.1) is regular, if

t ↦ W_{π}(th) is strongly continuous for any h ∈ 𝔤.

From now on we assume that we are given a regular representation.

By the Stone theorem, for any h ∈ 𝔤 we can define the corresponding field operator

ϕ_{π}(h) := −i\frac{d}{dt}W_{π}(th)|_{t=0}.

The following proposition is well known (see eg [BR]).

Proposition 2.1 i) In the sense of a quadratic form on D(ϕ_{π}(h_{1})) ∩ D(ϕ_{π}(h_{2})) the Heisenberg commutation relations are satisfied:

[ϕ_{π}(h_{1}), ϕ_{π}(h_{2})] = iσ(h_{1}, h_{2}). (2.4)

ii) W_{π}(g) leaves invariant D(ϕ_{π}(h)) and

[ϕ_{π}(h), W_{π}(g)] = iσ(g, h)W_{π}(g). (2.5)

iii) Let 𝔣 be a finite dimensional subspace of 𝔤. Then

𝔣 ∋ f ↦ W_{π}(h + f)

is strongly continuous for any h ∈ 𝔤. iv) If 𝔣 is a finite dimensional subspace of 𝔤, then the intersection of D(ϕ_{π}(h_{p}) ⋯ ϕ_{π}(h_{1})), h_{1}, …, h_{p} ∈ 𝔣, p ∈ IN is dense in H.

2.3 Creation and annihilation operators

From now on we assume that 𝔤 is equipped with a complex structure (that is an IR−linear operator i : 𝔤 → 𝔤 with i^{2} = −1). We assume that σ and i are compatible in the following sense:

\begin{matrix}σ(ih_{1}, h_{2}) + σ(h_{1}, ih_{2}) = 0, \\ σ(h, ih) > 0, h ≠ 0. \end{matrix}

(In particular, this forces σ to be non-degenerate). Then

(h_{1}|h_{2}) := σ(h_{1}, ih_{2}) + iσ(h_{1}, h_{2})

defines a positive definite scalar product. From now on we will treat 𝔤 as a complex space equipped with this scalar product. One defines the creation and annihilation operators as

\begin{matrix}a_{π}^{∗}(h) = \frac{1}{\sqrt{2}}(ϕ_{π}(h) − iϕ_{π}(ih)), \\ a_{π}(h) = \frac{1}{\sqrt{2}}(ϕ_{π}(h) + iϕ_{π}(ih)). \end{matrix} (2.6)

Clearly,

ϕ_{π}(h) := \frac{1}{\sqrt{2}}(a_{π}^{∗}(h) + a_{π}(h)), h ∈ 𝔤. (2.7)

Proposition 2.2 i) The operators a_{π}^{∗}(h) and a_{π}(h) with domain D(ϕ_{π}(h)) ∩ D(ϕ_{π}(ih)) are closed. (By Proposition 2.1 iii) this domain is dense in H). ii) The commutation relations are true in the sense of a quadratic form:

\begin{matrix}[a_{π}(h_{1}), a_{π}^{∗}(h_{2})] = (h_{1}|h_{2})1l, \\ [a_{π}(h_{2}), a_{π}(h_{1})] = [a_{π}^{∗}(h_{2}), a^{∗}(h_{1})] = 0. \end{matrix} (2.8)

iii) W_{π}(g) leaves invariant D(a_{π}^{#}(h)) and

\begin{matrix}[a_{π}(h), W_{π}(g)] = \frac{i}{\sqrt{2}} \overline{(g, h)}W_{π}(g), \\ [a_{π}^{∗}(h), W_{π}(g)] = −\frac{i}{\sqrt{2}}(g, h)W_{π}(g). \end{matrix} (2.9)

iv) If 𝔣 is a finite dimensional subspace of 𝔤, then the intersection of D(a_{π}^{#}(h_{p}) ⋯ a_{π}^{#}(h_{1})), h_{1}, …, h_{p} ∈ 𝔣, p ∈ IN is dense in H.

3 Operators in bosonic Fock spaces

We recall various constructions on bosonic Fock spaces.

3.1 Bosonic Fock spaces

Let 𝔥 be a Hilbert space, which we will call the 1-particle space. Let ⊗_{s}^{n}𝔥 denote the symmetric nth tensor power of 𝔥. Let S_{n} denote the orthogonal projection of ⊗^{n}𝔥 onto ⊗_{s}^{n}𝔥. If u ∈ ⊗_{s}^{p}𝔥 and v ∈ ⊗_{s}^{q}𝔥, then we will write

u ⊗_{s} v := S_{p+q}u ⊗ v ∈ ⊗_{s}^{p+q}𝔥.

If a ∈ B(⊗_{s}^{p}𝔥, ⊗_{s}^{r}𝔥) and b ∈ B(⊗_{s}^{q}𝔥, ⊗_{s}^{s}𝔥), then we will write

a ⊗_{s} b := S_{r+s}a ⊗ b ∈ B(⊗_{s}^{p+q}𝔥, ⊗_{s}^{r+s}𝔥).

We define the bosonic Fock space over 𝔥 to be the direct sum

Γ(𝔥) := ⨁_{n=0}^{∞} ⊗_{s}^{n}𝔥.

Ω will denote the vacuum vector – the vector 1 ∈ C = ⊗_{s}^{0}𝔥. The number operator N is defined as

N|_{⨂_{s}^{n} 𝔥} = n1l.

For a selfadjoint operator A on Γ(𝔥), we denote by H_{comp}(A) the space

H_{comp}(A) = {u ∈ H |χ(A)u = u, χ ∈ C_{0}^{∞}(IR)}.

We define the space of finite particle vectors and finite particle operators:

Γ_{fin}(𝔥) = H_{comp}(N) := {u ∈ Γ(𝔥) | for some n ∈ IN, 1l_{[0,n]}(N)u = u},
B_{fin}(Γ(𝔥)) := {B ∈ B(Γ(𝔥)) | for some n ∈ IN, 1l_{[0,n]}(N)B1l_{[0,n]}(N) = B}.

3.2 Creation and annihilation operators

There exists a natural representation of CCR over 𝔥 in Γ(𝔥) (where 𝔥 is equipped with the symplectic form Im(·|·)). To construct this representation it is natural to proceed in the reverse order from the one used in Sect. 2: first one constructs creation/annihilation operators, then field operators and then Weyl operators. If h ∈ 𝔥, we define the creation operator a^{∗}(h) by setting

\begin{matrix}a^{∗}(h) : Γ(𝔥) → Γ(𝔥), \\ a^{∗}(h)u := \sqrt{n + 1}h ⊗_{s} u, u ∈ ⊗_{s}^{n}𝔥. \end{matrix}

a(h) denotes the adjoint of a^{∗}(h), and is called the annihilation operator. Both a^{∗}(h) and a(h) are defined on Γ_{fin}(𝔥) and can be extended to densely defined closed operators on Γ(𝔥). By writing a^{♯}(h) we will mean both a^{∗}(h) and a(h).

Creation and annihilation operators a^{#}(h) on a Fock space satisfy (2.8) .

In our paper we will usually have

𝔥 = L^{2}(IR, dk). (3.1)

Then we will often write (as is customary in the literature)

a^{∗}(h) = ∫ a^{∗}(k)h(k)dk, a(h) = ∫ a(k)h̄(k)dk,

where a^{∗}(k) and a(k), k ∈ IR, have the meaning of operator valued distributions.

3.3 Field operators

We define the field operator

ϕ(h) := \frac{1}{\sqrt{2}}(a^{∗}(h) + a(h)), h ∈ 𝔥.

The operators ϕ(h) are essentially selfadjoint on Γ_{fin}(𝔥) and can be extended to self-adjoint operators on Γ(𝔥). Field operators ϕ(h) on a Fock space satisfy (2.4). In the case of (3.1), one can also write

ϕ(h) = ∫ h(k)ϕ(k)dk,

where ϕ(k) is an operator valued distribution

ϕ(k) := \frac{1}{\sqrt{2}}(a^{∗}(k) + a(k)), k ∈ IR.

3.4 Weyl operators

We introduce also the Weyl operators:

W (h) := e^{iϕ(h)}, h ∈ 𝔥.

The map 𝔥 ∋ h ↦ W (h) is a regular representation of CCR over 𝔥 in Γ(𝔥). Moreover, Weyl operators in a Fock space have the following properties: Proposition 3.1 i) the map

IR ∋ s ↦ W (sh)(N + 1)^{− \frac{1}{2}}

is C^{1} in the strong topology and the map

IR ∋ s ↦ W (sh)(N + 1)^{− \frac{1}{2} −ϵ}

is C^{1} in the norm topology. More precisely,

lim_{s→0} sup_{∥h∥≤C} s^{−1}∥(W (sh) − 1l − isϕ(h))(N + 1)^{−1/2−ϵ}∥ = 0.

ii)

∥(W (h_{1}) − W (h_{2}))u∥ ≤ C_{ϵ}∥h_{1} − h_{2}∥^{ϵ}((∥h_{1}∥^{2} + ∥h_{2}∥^{2})^{\frac{ϵ}{2}} ∥u∥ + ∥(N + 1)^{\frac{ϵ}{2}} u∥).

3.5 Operator dΓ

If b is an operator on 𝔥, we define the operator

dΓ(b) : Γ(𝔥) → Γ(𝔥),
dΓ(b)|_{⨂_{s}^{n} 𝔥} := ∑_{j=1}^{n} 1l^{⊗(j−1)} ⊗ b ⊗ 1l^{⊗(n−j)}
= nb ⊗_{s} 1l^{⊗(n−1)}.

An important example is the number operator N := dΓ(1).

Lemma 3.2 i) Heisenberg derivatives:

\frac{d}{dt}dΓ(b) = dΓ(\frac{d}{dt}b),

[dΓ(b_{1}), dΓ(b_{2})] = dΓ([b_{1}, b_{2}]).

ii) Commutation properties:

[dΓ(b), a^{∗}(h)] = a^{∗}(bh),

[dΓ(b), a(h)] = −a(b^{∗}h),
[dΓ(b), iϕ(h)] = ϕ(ibh), if b = b^{∗},
W (h)dΓ(b)W (−h) = dΓ(b) − ϕ(ibh) − \frac{1}{2}Re(bh|h) if b = b^{∗}.

iii) Estimates:

b_{1} ≤ b_{2} implies dΓ(b_{1}) ≤ dΓ(b_{2}),

∥N^{− \frac{1}{2}} dΓ(b)u∥ ≤ ∥dΓ(b^{∗}b)^{\frac{1}{2}} u∥,
dΓ(b)^{α} ≤ N^{1−α}dΓ(b^{α}), if b ≥ 0, 1 ≤ α,

dΓ(ab) ≤ dΓ(a^{p})^{\frac{1}{p}} dΓ(b^{q})^{\frac{1}{q}} , if a ≥ 0, b ≥ 0, ab = ba, p^{−1} + q^{−1} = 1.

3.6 Functor Γ

Let 𝔥_{i}, i = 1, 2 be Hilbert spaces. Let q : 𝔥_{1} ↦ 𝔥_{2} be a bounded linear operator. We define

\begin{matrix}Γ(q) : Γ(𝔥_{1}) ↦ Γ(𝔥_{2}) \\ Γ(q)|_{⨂_{s}^{n} 𝔥_{1}} = q ⊗ ⋯ ⊗ q. \end{matrix}

The Γ functor has the following properties: Lemma 3.3 i) Relationship with dΓ: assume 𝔥_{1} = 𝔥_{2}. Then

e^{dΓ(b)} = Γ(e^{b}).

ii) Intertwining properties:

\begin{matrix}Γ(q)a^{∗}(h_{1}) = a^{∗}(qh_{1})Γ(q), h_{1} ∈ 𝔥_{1}, \\ Γ(q)a(q^{∗}h_{2}) = a(h_{2})Γ(q), h_{2} ∈ 𝔥_{2}. \end{matrix}

If q is isometric, that is q^{∗}q = 1, then

\begin{matrix}Γ(q)a^{♯}(h_{1}) = a^{♯}(qh_{1})Γ(q), \\ Γ(q)ϕ(h_{1}) = ϕ(qh_{1})Γ(q). \end{matrix}

If q is unitary, then

\begin{matrix}Γ(q)a^{♯}(h_{1})Γ(q^{−1}) = a^{♯}(qh_{1}), \\ Γ(q)ϕ(h_{1})Γ(q^{−1}) = ϕ(qh_{1}). \end{matrix}

iii) If ∥q∥ ≤ 1, then

∥Γ(q)∥ = 1.

3.7 Operator dΓ(q, r)

Let q, r be operators from 𝔥_{1} to 𝔥_{2}. We define

dΓ(q, r) : Γ(𝔥_{1}) → Γ(𝔥_{2}),
dΓ(q, r)|_{⨂_{s}^{n} 𝔥_{1}} := ∑_{j=1}^{n} q^{⊗(j−1)} ⊗ r ⊗ q^{⊗(n−j)}
= nr ⊗_{s} q^{⊗(n−1)}.

Lemma 3.4 i) Relationship with dΓ and Γ:

dΓ(1, r) = dΓ(r), dΓ(r, r) = NΓ(r).

ii) Heisenberg derivatives of Γ(q):

dΓ(b_{2})Γ(q) = dΓ(q, b_{2}q), Γ(q)dΓ(b_{1}) = dΓ(q, qb_{1}),
\frac{d}{dt}Γ(q) = dΓ(q, \frac{d}{dt}q).

iii) Intertwining properties:

a(h_{2})dΓ(q, r) = dΓ(q, r)a(q^{∗}h_{2}) + Γ(q)a(r^{∗}h_{2}),
dΓ(q, r)a^{∗}(h_{1}) = a^{∗}(qh_{1})dΓ(q, r) + a^{∗}(rh_{1})Γ(q).

iv) Estimates:

0 ≤ r, and 0 ≤ q ≤ 1 implies dΓ(q, r) ≤ dΓ(r),
|(u_{2}|dΓ(q, r_{2}r_{1})u_{1})| ≤ ∥dΓ(r_{2}r_{2}^{∗})^{\frac{1}{2}} u_{2}∥∥dΓ(r_{1}^{∗}r_{1})^{\frac{1}{2}} u_{1}∥, ∥q∥ ≤ 1,
∥N^{− \frac{1}{2}} dΓ(q, r)u∥ ≤ ∥dΓ(r^{∗}r)^{\frac{1}{2}} u∥, ∥q∥ ≤ 1.

3.8 Tensor product of Fock spaces

We will adopt the following convention for tensor products: E ⊗ F will denote the algebraic tensor product of E and F, except when E, F are both Hilbert spaces, in which case it will denote the hilbertian tensor product. Let 𝔥_{i}, i = 1, 2 be two Hilbert spaces. Let i_{1}, i_{2} the injections of 𝔥_{1}, 𝔥_{2} into 𝔥_{1} ⊕ 𝔥_{2}. We define U : Γ(𝔥_{1}) ⊗ Γ(𝔥_{2}) → Γ(𝔥_{1} ⊕ 𝔥_{2}) as follows:

Uu ⊗ v := \sqrt{\frac{(p + q)!}{p!q!}}Γ(i_{1})u ⊗_{s} Γ(i_{2})v, u ∈ ⊗_{s}^{p}𝔥_{1}, u ∈ ⊗_{s}^{q}𝔥_{2}. (3.1)

Proposition 3.5 i) U is unitary, ii) UΩ ⊗ Ω = Ω, iii)

\begin{matrix}a^{♯}(h_{1} ⊕ h_{2})U = U(a^{♯}(h_{1}) ⊗ 1l + 1l ⊗ a^{♯}(h_{2})), h_{1} ∈ 𝔥_{1}, h_{2} ∈ 𝔥_{2}, \\ ϕ(h_{1} ⊕ h_{2})U = U(ϕ(h_{1}) ⊗ 1l + 1l ⊗ ϕ(h_{2})), h_{1} ∈ 𝔥_{1}, h_{2} ∈ 𝔥_{2}. \end{matrix}

iv) (3.2) dΓ(b_{1} ⊕ b_{2})U = U(dΓ(b_{1}) ⊗ 1l + 1l ⊗ dΓ(b_{2})),

UΓ(q_{1}) ⊗ Γ(q_{2}) = Γ(q_{1} ⊕ q_{2})U.

3.9 Scattering identification operator I

Along with the space Γ(𝔥) we will consider the space Γ(𝔥 ⊕ 𝔥) ≃ Γ(𝔥) ⊗ Γ(𝔥). We will use the notation

N_{0} := N ⊗ 1l, N_{∞} := 1l ⊗ N.

We will also write

a_{0}^{♯}(h) := a^{♯}(h) ⊗ 1l, a_{∞}^{♯}(h) := 1l ⊗ a^{♯}(h).

Following [HuSp1], we define the scattering identification operator

I : Γ_{fin}(𝔥) ⊗ Γ_{fin}(𝔥) → Γ_{fin}(𝔥),

Iu ⊗ v := \sqrt{\frac{(p + q)!}{p!q!}}u ⊗_{s} v, u ∈ ⊗_{s}^{p}𝔥, v ∈ ⊗_{s}^{q}𝔥. (3.3)

formula (3.1)

Another formula defining I is

I := Γ(i)U

where U : Γ(𝔥) ⊗ Γ(𝔥) → Γ(𝔥 ⊕ 𝔥) is the unitary operator introduced in (3.1) for 𝔥_{1} = 𝔥_{2} = 𝔥 and

\begin{matrix}i : 𝔥 ⊕ 𝔥 → 𝔥, \\ (h_{0}, h_{∞}) ↦ h_{0} + h_{∞}. \end{matrix}

Note that since ∥i∥ = \sqrt{2}, the operator Γ(i) is unbounded. Therefore, I is unbounded too. Yet another formula defining I is:

I Π_{i=1}^{n} a^{∗}(h_{i})Ω ⊗ Π_{i=1}^{p} a^{∗}(g_{i})Ω := Π_{i=1}^{p} a^{∗}(g_{i}) Π_{i=1}^{n} a^{∗}(h_{i})Ω, h_{i}, g_{i} ∈ 𝔥. (3.4)

If 𝔥 = L^{2}(IR, dk), then we can write still another formula for I:

Iu ⊗ ψ = \frac{1}{(p!)^{\frac{1}{2}}} ∫ ψ(k_{1}, ⋯, k_{p})a^{∗}(k_{1}) ⋯ a^{∗}(k_{p})udk_{1} ⋯ dk_{r}, u ∈ Γ(𝔥), ψ ∈ ⊗_{s}^{p}𝔥. (3.5)

Proposition 3.6 i) Let b, q be operators on 𝔥. Then dΓ(b)I = I(dΓ(b) ⊗ 1l + 1l ⊗ dΓ(b)),

Γ(q)I = IΓ(q) ⊗ Γ(q).

ii) For h ∈ 𝔥

\begin{matrix}a(h)I = I(a_{0}(h) + a_{∞}(h)), \\ a^{∗}(h)I = Ia_{0}^{∗}(h) = Ia_{∞}^{∗}(h). \end{matrix}

iii) (3.6) I(N_{0} + 1)^{−k/2}1l_{[0,k]}(N_{∞}) is bounded.

3.10 Operator I(j)

Let j_{0}, j_{∞} be two operators on 𝔥. Set j = (j_{0}, j_{∞}). We define

I(j) : Γ_{fin}(𝔥) ⊗ Γ_{fin}(𝔥) → Γ_{fin}(𝔥) I(j) := IΓ(j_{0}) ⊗ Γ(j_{∞}).

If we identify j with the operator

\begin{matrix}j : 𝔥 ⊕ 𝔥 → 𝔥, \\ j(h_{0} ⊕ h_{∞}) := j_{0}h_{0} + j_{∞}h_{∞}, \end{matrix} (3.7)

then we have

I(j) = Γ(j)U.

Remark 3.7 I(j) equals Γ̌(j^{∗})^{∗} in the notation of [DG1].

Note that I = I(1, 1). Other formulas defining I(j) are

I(j) Π_{i=1}^{n} a^{∗}(h_{i})Ω ⊗ Π_{i=1}^{p} a^{∗}(g_{i})Ω := Π_{i=1}^{p} a^{∗}(j_{0}g_{i}) Π_{i=1}^{n} a^{∗}(j_{∞}h_{i})Ω, h_{i}, g_{i} ∈ 𝔥, (3.8)

I^{∗}(j)Π_{i=1}^{n}a^{∗}(h_{i})Ω := Π_{i=1}^{n} (a_{0}^{∗}(j_{0}^{∗}h_{i}) + a_{∞}^{∗}(j_{∞}^{∗}h_{i})) Ω ⊗ Ω, h_{i} ∈ 𝔥. (3.9) Lemma 3.8 i)

I(j̃)I^{∗}(j) = Γ(j̃_{0}j_{0}^{∗} + j̃_{∞}j_{∞}^{∗}).

In particular, if j_{0}^{∗} + j_{∞}^{∗} = 1l, then

II^{∗}(j) = 1l.

ii) Intertwining properties: For h ∈ 𝔥 a(h)I(j) = I(j)(a_{0}(j_{0}^{∗}h) + a_{∞}(j_{∞}^{∗}h)),

a^{∗}(j_{0}h)I(j) = I(j)a_{0}^{∗}(h),
a^{∗}(j_{∞}h)I(j) = I(j)a_{∞}^{∗}(h).

iv) I(j) is bounded iff ∥j_{0}^{∗}j_{0} + j_{∞}^{∗}j_{∞}∥ ≤ 1, and then ∥I(j)∥ = 1.

Let us note some additional properties of I(j) in the coisometric case.

Lemma 3.9 Assume j_{0}j_{0}^{∗} + j_{∞}j_{∞}^{∗} = 1. (3.10)

(This assumption implies that j is coisometric, that is jj^{∗} = 1). Then i)

I(j)I^{∗}(j) = 1l.

ii) Intertwining properties:

a^{#}(h)I(j) = I(j) (a_{0}^{#}(j_{0}^{∗}h) + a_{∞}^{#}(j_{∞}^{∗}h)) ,

ϕ(h)I(j) = I(j) (ϕ_{0}(j_{0}^{∗}h) + ϕ_{∞}(j_{∞}^{∗}h)) .

iii) If in addition j_{0}, j_{∞} are self-adjoint, then

dΓ(b) = I(j) (dΓ(b) ⊗ 1l + 1l ⊗ dΓ(b)) I^{∗}(j) + \frac{1}{2}dΓ(ad_{j_{0}}^{2}b + ad_{j_{∞}}^{2}b).

3.11 Operator dI(j, k)

Let j = (j_{0}, j_{∞}), k = (k_{0}, k_{∞}) be pairs of maps from 𝔥 to 𝔥. We define

dI(j, k) : Γ_{fin}(𝔥) ⊗ Γ_{fin}(𝔥) → Γ_{fin}(𝔥)

as follows:

dI(j, k) := I(dΓ(j_{0}, k_{0}) ⊗ Γ(j_{∞}) + Γ(j_{0}) ⊗ dΓ(j_{∞}, k_{∞})).

Equivalently, treating j and k as maps from 𝔥 ⊕ 𝔥 to 𝔥, as in (3.7), we can write

dI(j, k) := dΓ(j, k)U.

Remark 3.10 dI(j, k) equals dΓ̌(j^{∗}, k^{∗})^{∗} in the notation of [DG1].

Lemma 3.11 i) Heisenberg derivative of I(j):

\frac{d}{dt}I(j) = dI(j, \frac{d}{dt}j),

I(j) (dΓ(b_{0}) ⊗ 1l + 1l ⊗ dΓ(b_{∞})) = dI(j, k),

dΓ(b)I(j) = dI(j, bj).

Here b, b_{0}, b_{∞} are operators on 𝔥 and k = (j_{0}b_{0}, j_{∞}b_{∞}). ii) Intertwining properties:

\begin{matrix}dI(j, k)a_{0}^{∗}(h) = a^{∗}(j_{0}h)dI(j, k) + a^{∗}(k_{0}h)I(j), \\ dI(j, k)(a_{0}(j_{0}^{∗}h) + a_{∞}(j_{∞}^{∗}h)) + I(j)(a_{0}(k_{0}^{∗}h) + a_{∞}(k_{∞}^{∗}h)) = a(h)dI(j, k). \end{matrix}

iv) If j_{0}j_{0}^{∗} + j_{∞}j_{∞}^{∗} ≤ 1, k_{0}, k_{∞} are self-adjoint, we have the estimate:

|(u_{2}|dI^{∗}(j, k)u_{1})| ≤ ∥dΓ(|k_{0}|)^{\frac{1}{2}} ⊗ 1lu_{2}∥∥dΓ(|k_{0}|)^{\frac{1}{2}} u_{1}∥
+∥1l ⊗ dΓ(|k_{∞}|)^{\frac{1}{2}} u_{2}∥∥dΓ(|k_{∞}|)^{\frac{1}{2}} u_{1}∥, u_{1} ∈ Γ(𝔥), u_{2} ∈ Γ(𝔥) ⊗ Γ(𝔥).

v) If j_{0}j_{0}^{∗} + j_{∞}j_{∞}^{∗} ≤ 1, then

∥(N_{0} + N_{∞})^{− \frac{1}{2}} dI^{∗}(j, k)u∥ ≤ ∥dΓ(k_{0}k_{0}^{∗} + k_{∞}k_{∞}^{∗})^{\frac{1}{2}} u∥, u ∈ Γ(𝔥).

3.12 Wick polynomials

Let w ∈ B(⊗_{s}^{p}𝔥, ⊗_{s}^{q}𝔥). We define the operator

Wick(w) : Γ_{fin}(𝔥) → Γ_{fin}(𝔥)

as follows:

Wick(w)|_{⨂_{s}^{n} 𝔥} := \frac{\sqrt{n!(n + q − p)!}}{(n − p)!}w ⊗_{s} 1l^{⊗(n−p)}. (3.11)

This definition extends to w ∈ B_{fin}(Γ(𝔥)) by linearity. The operator Wick(w) is called a Wick polynomial. The operator w is called the symbol of the Wick polynomial Wick(w).

Before we describe properties of Wick polynomials, let us introduce more definitions. If u is an element of a Hilbert space H, we denote by (u| the map H ∋ v ↦ (u, v) ∈ C and by |u) : C → H its adjoint.

If u ∈ ⊗_{s}^{m}𝔥, v ∈ ⊗_{s}^{n}𝔥, w ∈ B(⊗_{s}^{p}𝔥, ⊗_{s}^{q}𝔥) with m ≤ p, n ≤ q, then, to simplify the notation, we will introduce the ‘contracted’ symbols

(v|w := ((v| ⊗_{s} 1l^{⊗(q−n)}) w ∈ B(⊗_{s}^{p}𝔥, ⊗_{s}^{q−n}𝔥),
w|u) := w (|u) ⊗_{s} 1l^{⊗(p−m)}) ∈ B(⊗_{s}^{p−m}𝔥, ⊗_{s}^{q}𝔥),
(v|w|u) := ((v| ⊗_{s} 1l^{⊗(q−n)}) w (|u) ⊗_{s} 1l^{⊗(p−m)}) ∈ B(⊗_{s}^{p−m}𝔥, ⊗_{s}^{q−n}𝔥).

Theorem 3.12 i) Case p = q = 0. Let λ ∈ C = B(⊗_{s}^{0}𝔥, ⊗_{s}^{0}h). Then Wick(λ) = λ1l. ii) Case p = q = 1. (Note that ⊗_{s}^{1}𝔥 = 𝔥) For b ∈ B(𝔥) we have

Wick(b) = dΓ(b).

iii) Cases q = 0, p = 1 and q = 1, p = 0. For h ∈ 𝔥 we have

Wick(|h)) = a^{∗}(h), Wick((h|) = a(h).

iv) Let u ∈ ⊗_{s}^{m}𝔥, v ∈ ⊗_{s}^{n}𝔥, w ∈ B(⊗_{s}^{p}𝔥, ⊗_{s}^{q}𝔥). Then

Wick(|v) ⊗_{s} w ⊗_{s} (u|) = Wick(|v))Wick(w)Wick((u|).

v) Let h′_{1}, …, h′_{p}, h_{1}, …, h_{q} ∈ 𝔥. Then

Wick (| h_{1} ⊗_{s} ⋯ ⊗_{s} h_{q})(h′_{p} ⊗_{s} ⋯ ⊗_{s} h′_{1})|) = a^{∗}(h_{1}) ⋯ a^{∗}(h_{q})a(h′_{p}) ⋯ a(h′_{1}).

vi) Let w ∈ B(⊗_{s}^{p}𝔥, ⊗_{s}^{q}𝔥). Then

Wick(w)^{∗} = Wick(w^{∗}).

Note that v) of the above theorem (which follows immediately from iv)), justifies the name Wick polynomials.

If we fix a basis {h_{i}}_{i∈I} of 𝔥, then any operator in B(⊗_{s}^{p}𝔥, ⊗_{s}^{q}𝔥) can be written as a sum (convergent for the weak topology):

w = ∑_{i_{1},...,i_{q},i′_{1},...,i′_{p}} w_{i_{1},...,i_{q},i′_{p},...,i′_{1}}|h_{i_{1}} ⊗_{s} ⋯ ⊗_{s} h_{i_{q}})(h_{i′_{p}} ⊗_{s} ⋯ ⊗_{s} h_{i′_{1}}|

where we can assume that w_{i_{1},...,i_{q},i′_{p},...,i′_{1}} is separately symmetric wrt the first q and the last p indices. Then, writing a_{i}^{#} for a^{#}(h_{i}), we have

Wick(w) = ∑_{i_{1},...,i_{q},i′_{p},...,i′_{1}} w_{i_{1},...,i_{q},i′_{p},...,i′_{1}}a_{i_{1}}^{∗} ⋯ a_{i_{q}}^{∗}a_{i′_{p}} ⋯ a_{i′_{1}}.

In later sections, we will consider the case when 𝔥 = L^{2}(IR, dk). Any operator w from S(IR^{p}) to S′(IR^{q}), in particular, any w ∈ B(⊗_{s}^{p}𝔥, ⊗_{s}^{q}𝔥), has a distributional kernel

w(k_{1}, …, k_{q}, k′_{p}, …, k′_{1}) ∈ S′(IR^{p+q}), (3.12)

where we can assume that the kernel w in (3.12) is separately symmetric wrt the first q and the last p variables. The following formal expression is then commonly used to denote the Wick polynomial Wick(w):

∫ w(k_{1}, …, k_{q}, k′_{p}, …, k′_{1})a^{∗}(k_{1}) ⋯ a^{∗}(k_{q})a(k′_{p}) ⋯ a(k′_{1})dk_{1} ⋯ dk_{q}dk′_{p} ⋯ dk′_{1}. (3.13)

Although the definition (3.11) always makes sense, let us describe a few cases in which a rigorous meaning can be attached to the formal expression (3.13).

First of all if u ∈ Γ_{fin}(S(IR)), a(k_{1}) … a(k_{p})u is well defined as an element of S(IR^{p}) ⊗ Γ(𝔥). This shows that the expression

\begin{aligned}∫ & w(k_{1}, …, k_{q}, k′_{p}, …, k′_{1}) \\ & × (a(k_{1}) … a(k_{q})u|a(k′_{1}) … a(k_{p})u)_{Γ(𝔥)} dk_{1}, ⋯ dk_{q}dk′_{p} ⋯ dk′_{1} \end{aligned} (3.14)

is well defined for u ∈ Γ_{fin}(S(IR)), w ∈ S′(IR^{p+q}). Hence if w ∈ S′(IR^{p+q}), (3.13) always makes sense as a quadratic form on Γ_{fin}(S(IR)).

If u ∈ D(N^{p/2}), a(k_{1}) … a(k_{p})u is also well defined as an element of ⨂_{s}^{p} L^{2}(IR) ⊗ Γ(L^{2}(IR)). and the expression (3.14) is well defined for u ∈ D(N^{sup(p,q)/2}), w ∈ L^{2}(IR^{p+q}). Hence if w ∈ L^{2}(IR^{p+q}), (3.12) makes sense as a quadratic form on D(N^{sup(p,q)/2}).

The following proposition summarizes basic properties of Wick polynomials.

Proposition 3.13 i) If w ∈ B(⊗_{s}^{p}𝔥, ⊗_{s}^{q}𝔥) and k + m ≥ \frac{p+q}{2} , then

∥(N + 1)^{−k}Wick(w)(N + 1)^{−m}∥ ≤ ∥w∥.

If moreover, s- lim w_{n} = w, then

s- lim_{n→∞}(N + 1)^{−k}Wick(w_{n})(N + 1)^{−m} = (N + 1)^{−k}Wick(w)(N + 1)^{−m}.

ii) Identities:

Let w ∈ B_{fin}(Γ(𝔥)), b ∈ B(𝔥). Then

(3.15) [dΓ(b), Wick(w)] = Wick([dΓ(b), w]).

Let w ∈ B_{fin}(Γ(𝔥_{2}), Γ(𝔥_{1})), q ∈ B(𝔥_{1}, 𝔥_{2}). Then

(3.16) Γ(q)Wick(wΓ(q)) = Wick(Γ(q)w)Γ(q).

Let w ∈ B_{fin}(Γ(𝔥_{1}), Γ(𝔥_{1})), q ∈ B(𝔥_{1}, 𝔥_{2}). Then

(3.17) Γ(q)Wick(w) = Wick(Γ(q)wΓ(q^{∗}))Γ(q), for isometric q,

(3.18) Γ(q)Wick(w)Γ(q^{−1}) = Wick(Γ(q)wΓ(q^{−1})), for unitary q.

Let w ∈ B_{fin}(Γ(𝔥)), h ∈ 𝔥. Then

(3.19) [Wick(w), a^{∗}(h)] = pWick(w|h)), [Wick(w), a(h)] = qWick((h|w),

(3.20) W (h)Wick(w)W (−h) = ∑_{s=0}^{p} ∑_{r=0}^{q} \frac{p!}{s!} \frac{q!}{r!}(\frac{i}{\sqrt{2}})^{p+q−r−s}Wick(w_{s,r}),

where w_{s,r} = (h^{⊗(q−r)}|w|h^{⊗(p−s)}). (3.21)

Proof. The first part of i) is a particular case of the well known N_{τ} estimates (see eg [GJ1]). It follows directly from the definition (3.11) and the fact that

\frac{(n!(n + q − p)!)^{\frac{1}{2}}}{n − p!} ≤ (n + q − p)^{q/2}n^{p/2}.

The second part of i) follows similarly from (3.11). All identities of ii) are easy, except for the last one which follows for example from Thm. 3.12 v) and the identity

W (h)a^{∗}(f)W (−h) = a^{∗}(f) + \frac{i}{\sqrt{2}}(h|f).

□

Let now K be an additional Hilbert space. If w ∈ B_{fin}(K ⊗ Γ(𝔥)), then we can also define Wick(w) as an operator acting on K ⊗ Γ_{fin}(𝔥), by

Wick(w)|_{K⊗⊗_{s}^{n}𝔥} := \frac{\sqrt{n!(n + q − p)!}}{(n − p)!}1l_{K} ⊗ S_{n+q−p}w ⊗ 1l^{⊗(n−p)},

for w ∈ B(K ⊗ ⊗_{s}^{p}𝔥, K ⊗ ⊗_{s}^{q}𝔥). This construction can be used for example if one considers generalizations of Pauli-Fierz Hamiltonians with more general interactions than those considered in [DG1].

In particular, this additional space can be also a Fock space. Then if

w ∈ B(⊗_{s}^{p_{1}}𝔥_{1} ⊗ ⊗_{s}^{p_{2}}𝔥_{2}, ⊗_{s}^{q_{1}}𝔥_{1} ⊗ ⊗_{s}^{q_{2}}𝔥_{2}),

we define:

Wick ⊗ Wick(w)|_{⊗_{s}^{n_{1}} 𝔥_{1}⊗⊗_{s}^{n_{2}} 𝔥_{2}}
:= ^{\frac{\sqrt{n_{1}!(n_{1}+q_{1}−p_{1})!}}{(n_{1}−p_{1})!}} ^{\frac{\sqrt{n_{2}!(n_{2}+q_{2}−p_{2})!}}{(n_{2}−p_{2})!}}(S_{n_{1}+q_{1}−p_{1}} ⊗ S_{n_{2}+q_{2}−p_{2}})1l^{⊗(n_{1}−p_{1})} ⊗ w ⊗ 1l^{⊗(n_{2}−p_{2})}.

We extend this definition to w ∈ B(Γ_{fin}(𝔥_{1}) ⊗ Γ_{fin}(𝔥_{2})) by linearity. The following proposition is completely similar to Prop. 3.13.

Proposition 3.14 i) For w_{i} ∈ B(Γ_{fin}(𝔥_{i})), i = 1, 2 then

Wick ⊗ Wick(w_{1} ⊗ w_{2}) = Wick(w_{1}) ⊗ Wick(w_{2}).

ii) If

w = | ⊗_{s}_{1}^{q_{1}} h_{i,1})(⊗_{s}_{1}^{p_{1}} g_{i,1}| ⊗ | ⊗_{s}_{1}^{q_{2}} h_{i,2})(⊗_{s}_{1}^{p_{2}} g_{i,2}|

then

Wick ⊗ Wick(w) = (Π_{1}^{q_{1}}a^{∗}(h_{i,1}) ⊗ Π_{1}^{q_{2}}a^{∗}(h_{i,2}))(Π_{1}^{p_{1}}a(g_{i,1}) ⊗ Π_{1}^{p_{2}}a(g_{i,2})).

iii) For j_{i} ∈ B(𝔥_{i}), i = 1, 2, w ∈ B(Γ_{fin}(𝔥_{1}) ⊗ Γ_{fin}(𝔥_{2}))

Γ(j_{1}) ⊗ Γ(j_{2})Wick ⊗ Wick(wΓ(j_{1}) ⊗ Γ(j_{2})) = Wick ⊗ Wick(Γ(j_{1}) ⊗ Γ(j_{2})w)Γ(j_{1}) ⊗ Γ(j_{2}).

iv) for w ∈ B(⨂_{s}^{p_{1}} 𝔥_{1} ⊗ ⨂_{s}^{p_{2}} 𝔥_{2}, ⨂_{s}^{q_{1}} 𝔥_{1} ⊗ ⨂_{s}^{q_{2}} 𝔥_{2}) and k_{i} + m_{i} ≥ \frac{p_{i}+q_{i}}{2} , i = 1, 2 we have:

∥(N + 1)^{−k_{1}} ⊗ (N + 1)^{−k_{2}}Wick ⊗ Wick(w)(N + 1)^{−m_{1}} ⊗ (N + 1)^{−m_{2}}∥ ≤ ∥w∥.

In the next proposition we describe additional properties of Wick polynomials which have a nice formulation if one uses the Wick ⊗ Wick notation.

Proposition 3.15 i) Let w ∈ B_{fin}(Γ(𝔥_{1} ⊕ 𝔥_{2})). Then

U^{∗}Wick(w)U = Wick ⊗ Wick(Ũ^{∗}wŨ).

Here the map Ũ : Γ_{fin}(𝔥_{1}) ⊗ Γ_{fin}(𝔥_{2}) → Γ_{fin}(𝔥_{1} ⊕ 𝔥_{2}) is defined as follows (recall that i_{1}, i_{2} are the injections of 𝔥_{1}, 𝔥_{2} into 𝔥_{1} ⊕ 𝔥_{2}):

Ũu_{1} ⊗ u_{2} := \frac{(p + q)!}{p!q!}Γ(i_{1})u_{1} ⊗_{s} Γ(i_{2})u_{2}, u_{1} ∈ ⊗_{s}^{p}𝔥_{1}, u_{2} ∈ ⊗_{s}^{q}𝔥_{2}. (3.22)

ii) Let w ∈ B_{fin}(Γ(𝔥)). Then

Wick(w)I = IWick ⊗ Wick(P wĨ).

Here Ĩ: Γ_{fin}(𝔥) ⊗ Γ_{fin}(𝔥) → Γ_{fin}(𝔥) is defined as follows:

Ĩu ⊗ v := \frac{(p + q)!}{p!q!}u ⊗_{s} v, u ∈ ⊗_{s}^{p}𝔥, v ∈ ⊗_{s}^{q}𝔥. (3.23)

and P : Γ_{fin}(𝔥) → Γ_{fin}(𝔥) ⊗ Γ_{fin}(𝔥) is defined as

P u := u ⊗ Ω, u ∈ Γ_{fin}(𝔥).

iii) Let us keep the notation of ii) and let j be as in Subsection 3.10. Then

Wick(Γ(j_{0})w)I(j) = I(j)Wick ⊗ Wick(P wĨΓ(j_{0}) ⊗ Γ(j_{∞})).

Proof. By linearity it suffices to check the identities i) and ii) for w of rank one. One can then use the identities in Lemma 3.5 iii), Thm. 3.12 v) and Prop. 3.14 ii) to verify i) and ii). Finally iii) follows from ii), the fact that I(j) = IΓ(j_{0}) ⊗ Γ(j_{∞}) and Prop. 3.14 iii).□

In the following sections, we will need to estimate some commutators between Wick polynomials and Γ(q), I^{∗}(j) operators. To this end we will need the identities described in the next proposition.

Proposition 3.16 Let w ∈ B_{fin}(Γ(𝔥)). Then i) for q ∈ B(𝔥):

[Γ(q), Wick(w)] = Γ(q)Wick(w(1l − Γ(q))) + Wick((Γ(q) − 1l)w)Γ(q).

ii) for j = (j_{0}, j_{∞}) ∈ B(𝔥 ⊗ 𝔥, 𝔥):

I^{∗}(j)Wick(w) − (Wick(w) ⊗ 1l)I^{∗}(j)
= I^{∗}(j)Wick(w(1l − Γ(j_{0}^{∗})))
+ Wick ⊗ Wick((Γ(j_{0}^{∗}) − 1l) ⊗ Γ(j_{∞}^{∗})Ĩ^{∗}wP^{∗})I^{∗}(j)
+ Wick ⊗ Wick(1l ⊗ (Γ(j_{∞}^{∗}) − |Ω)(Ω|)Ĩ^{∗}wP^{∗})I^{∗}(j).

Proof. i) follows directly from the identity (3.16). To prove ii), we deduce from Prop. 3.15 that

I^{∗}(j)Wick(w) = I^{∗}(j)Wick(w(1l − Γ(j_{0}^{∗}))) + Wick ⊗ Wick(Γ(j_{0}^{∗}) ⊗ Γ(j_{∞}^{∗})Ĩ^{∗}wP^{∗})I^{∗}(j).

We use then the identities

w ⊗ |Ω)(Ω| = P wP^{∗}, P = (1l ⊗ |Ω)(Ω|)Ĩ^{∗},

to obtain

Wick(w) ⊗ 1l = Wick ⊗ Wick(w ⊗ |Ω)(Ω|)
= Wick ⊗ Wick((1l ⊗ |Ω)(Ω|)Ĩ^{∗}wP^{∗}).

Next we write

Γ(j_{0}^{∗}) ⊗ Γ(j_{∞}^{∗}) − 1l ⊗ |Ω)(Ω|
= (Γ(j_{0}^{∗}) − 1l) ⊗ Γ(j_{∞}^{∗}) + 1l ⊗ (Γ(j_{∞}^{∗}) − |Ω)(Ω|)

to obtain ii). □

Lemma 3.17 Assume that 𝔥 = L^{2}(IR, dk) and that w ∈ B(⊗_{s}^{p}𝔥, ⊗_{s}^{r}𝔥) is given by a kernel w as in (3.12) with w ∈ L^{2}(IR^{p+r}).

i) Let q ∈ B(𝔥), ∥q∥ ≤ 1. Then for m + k ≥ \frac{p+r}{2} :

∥(N + 1)^{−m}[Γ(q), Wick(w)](N + 1)^{−k}∥
≤ C_{p,r} sup_{1≤i≤p+r} ∥1l^{⊗(i−1)} ⊗ (1l − q) ⊗ 1l^{⊗(p+r−i)}w∥_{L^{2}(IR^{p+r})}.

(3.24)

ii) Let j = (j_{0}, j_{∞}) with j_{0}, j_{∞} ∈ B(𝔥), ∥j_{0}^{∗}j_{0} + j_{∞}^{∗}j_{∞}∥ ≤ 1. Then for m + k ≥ \frac{p+r}{2} :

∥(N_{0} + N_{∞} + 1)^{−m}(I^{∗}(j)Wick(w) − (Wick(w) ⊗ 1l)I^{∗}(j))(N + 1)^{−k}∥
≤ C_{p,r} sup_{1≤i≤p+r} ∥1l^{⊗(i−1)} ⊗ (1l − j_{0}) ⊗ 1l^{⊗(p+r−i)}w∥_{L^{2}(IR^{p+r})}
+ C_{p,r} sup_{1≤i≤p+r} ∥1l^{⊗(i−1)} ⊗ (j_{∞}) ⊗ 1l^{⊗(p+r−i)}w∥_{L^{2}(IR^{p+r})}.

(3.25)

Proof. To prove i) it suffices by Prop. 3.13 i) to estimate the operator norm of the symbols w(1l − Γ(q)) and (Γ(q) − 1l)w, which are bounded by the r.h.s. of (3.24).

Similarly to prove ii) it suffices by Prop. 3.14 iv) to estimate the operator norm of the symbols w(1l − Γ(j_{0}^{∗})), (Γ(j_{0}^{∗}) − 1l) ⊗ Γ(j_{∞}^{∗})Ĩ^{∗}wP^{∗} and (1l ⊗ (Γ(j_{∞}^{∗}) − |Ω)(Ω|)Ĩ^{∗}wP^{∗} (note that ∥I^{∗}(j)∥ = 1 by Lemma 3.8 iv) and I^{∗}(j)N = (N_{0} + N_{∞})I^{∗}(j)).

The norm of the first symbol is bounded by the r.h.s. of (3.25) by the same argument as in i). Since ∥Γ(j_{∞}^{∗})∥ = ∥P^{∗}∥ = 1 the norm of the second symbol is less than ∥wĨ(Γ(j_{0})1l) ⊗ 1l∥ which is bounded by the r.h.s. of (3.25). Similarly the norm of the third symbol is less than ∥wĨ1l⊗(Γ(j_{∞})−|Ω)(Ω|)∥. This is also bounded by the r.h.s. of (3.25). (Note that Γ(j_{∞})−|Ω)(Ω| vanishes on the vacuum sector). □

4 Fock representations of CCR

In this section we described the construction of the Fock subrepresentation of a regular CCR representation, (see [BR, CMR]).

4.1 Construction of the Fock subrepresentation

Suppose that we are given a regular CCR representation over a pre-Hilbert space 𝔤 in the Hilbert space H. Let 𝔥 be the completion of 𝔤.

We define the space of vacua

K_{π} := {u ∈ H | a_{π}(h)u = 0, h ∈ 𝔤}.

Proposition 4.1 i) K_{π} is a closed space. ii) K_{π} is contained in the set of analytic vectors of ϕ_{π}(h), h ∈ 𝔤.

Proof. K_{π} is closed as an intersection of null spaces of closed operators. To prove ii) we will show that (u|W_{π}(h)u) = exp(−∥h∥^{2}/4), u ∈ K_{π}. (4.1)

Clearly, K_{π} ⊂ D(ϕ_{π}(h)), hence for u ∈ K_{π}

f(t) := (u|W_{π}(th)u)

is continuously differentiable. We have

\frac{d}{dt}f(t) = i(u|ϕ_{π}(h)W_{π}(th)u)
= \frac{i}{\sqrt{2}}(a_{π}(h)u|W_{π}(th)u) + \frac{i}{\sqrt{2}}(u|W_{π}(th)a_{π}(h)u) − \frac{1}{2}t∥h∥^{2}(u|W_{π}(th)u)
= −\frac{1}{2}t∥h∥^{2}f(t).

Therefore

f(t) = exp(−t^{2}∥h∥^{2}/4),

which shows (4.1). Now the spectral theorem implies that u is an analytic vector for ϕ_{π}(h) and thus t ↦ W_{π}(th)u extends to an analytic function around the real axis. □

Define the space H_{π} := K_{π} ⊗ Γ(𝔥). We define Ω_{π} : K_{π} ⊗ Γ_{fin}(𝔤) → H by setting

Ω_{π}ψ ⊗ ϕ(h)^{p}Ω := ϕ_{π}(h)^{p}ψ, h ∈ 𝔤, ψ ∈ K^{+}. (4.2)

(Note that the vectors ϕ(h)^{p}Ω = 2^{−p/2}a^{∗}(h)^{p}Ω for h ∈ 𝔤 span ⊗_{s}^{p}𝔥)

Proposition 4.2 The map Ω_{π} extends to an isometric map

Ω_{π} : H_{π} → H,

satisfying Ω_{π}1l ⊗ a^{#}(h) = a_{π}^{#}(h)Ω_{π}, h ∈ 𝔤. (4.3)

4.2 Number operator

We discuss now the number operator N_{π} associated to a regular CCR representation in the space H. One can give two equivalent definitions of N_{π}. Note that a more general notion of a number operator associated to a CCR representation, (which in particular does not need to be a positive operator) has been introduced by Chaiken in [Ch1, Ch2].

The first definition uses the intertwiner Ω_{π}. Define D(N_{π}) := Ω_{π}K_{π} ⊗ D(N), which is a subspace of H_{π} whose closure is RanΩ_{π}. Let N_{π} be the operator on H with the domain D(N_{π}) defined by

N_{π} := Ω_{π}1l ⊗ NΩ_{π}^{∗}.

(Note that N_{π} needs not be densely defined).

Before we give an alternative definition of N_{π}, let us recall some facts about quadratic forms. We will assume that a positive quadratic form is defined on the whole space H and takes values in [0, ∞]. The domain of a positive quadratic form b is defined as

D(b) := {u ∈ H| b(u) < ∞}.

If the form b is closed, then there exists a unique positive self-adjoint operator B such that

D(b) = D(B^{\frac{1}{2}} ), b(u) = (u|Bu).

If A is a closed operator, then ∥Au∥^{2} is a closed form. The sum of closed forms is a closed form, and the supremum of a family of closed forms is a closed form.

The following theorem gives an alternative definition of N_{π}.

Theorem 4.3 For each finite dimensional space 𝔣 ⊂ 𝔤, one defines

n_{π,𝔣}(u) := ∑_{i=1}^{dim𝔣} ∥a_{π}(h_{i})u∥^{2},

where {h_{i}} is an orthonormal basis of 𝔣. (If u ∉ D(a_{π}(h_{i}) for some i, then n_{π,𝔣}(u) = ∞). Then the quadratic form n_{π,𝔣} does not depend on the choice of the basis {h_{i}} of 𝔣. The quadratic form n_{π} is defined by

n_{π}(u) := sup_{𝔣} n_{π,𝔣}(u), u ∈ H.

Then D(n_{π}) = D((N_{π})^{\frac{1}{2}} ) and

n_{π}(u) = (u|N_{π}u), u ∈ D(N_{π}).

In particular, RanΩ_{π} = \overline{D(n_{π})}.

To prepare for the proof of the above theorem, note that n_{π} defines a positive operator, which we denote temporarily Ñ_{π}, such that D(n_{π}) = D((Ñ_{π})^{\frac{1}{2}} ) and

n_{π}(u) = (u|Ñ_{π}u), u ∈ D(Ñ_{π}). (4.4)

Our aim is to show that Ñ_{π} = N_{π}.

Note also that D(n_{π}) ⊂ D(ϕ_{π}(h)), h ∈ 𝔤. (4.5)

Lemma 4.4 If v ∈ D(Ñπ^{\frac{1}{2}} ), and F is a Borel function, then

a_{π}(h)F (Ñ_{π} − 1)v = F (Ñ_{π})a_{π}(h)v. (4.6)

Proof. First we note that W_{π}(h) maps D(n_{π}) into itself and we have

n_{π}(W_{π}(h)u) = n_{π}(u) + (u|ϕ_{π}(ih)u) + ∥h∥^{2}∥u∥^{2}/2. (4.7)

In fact, using (2.5) we see that (4.7) is true if we replace n_{π} with n_{π,𝔣}, where 𝔣 is a finite subspace of 𝔤 containing h. Then (4.7) follows immediately.

By the polarization identity, (4.7) has the following consequence for u, w ∈ D(n_{π}):

(Ñπ^{\frac{1}{2}} W_{π}(h)w|Ñπ^{\frac{1}{2}} W_{π}(h)u) = (Ñπ^{\frac{1}{2}} w|Ñπ^{\frac{1}{2}} u) + (w|ϕ_{π}(ih)u) + ∥h∥^{2}(w|u)/2. (4.8)

Replacing w with W_{π}(h)^{∗}v and using the invariance of D(n_{π}) under W_{π}(h) we can rewrite (4.8) as follows, for u, v ∈ D(n_{π}):

\begin{matrix}(Ñπ^{\frac{1}{2}} v|Ñπ^{\frac{1}{2}} W_{π}(h)u) \\ = (Ñπ^{\frac{1}{2}} W_{π}(h)^{∗}v|Ñπ^{\frac{1}{2}} u) + (W_{π}(h)^{∗}v|ϕ_{π}(ih)u) + \frac{1}{2}∥h∥^{2}(W_{π}(h)^{∗}v|u). \end{matrix} (4.9)

Next assume in addition that u, v ∈ D(Ñ_{π}). Then we can rewrite (4.9) as

\begin{matrix}(Ñ_{π}v|W_{π}(h)u) \\ = (W_{π}(h)^{∗}v|Ñ_{π}u) + (W_{π}(h)^{∗}v|ϕ_{π}(ih)u) + \frac{1}{2}∥h∥^{2}(W_{π}(h)^{∗}v|u). \end{matrix} (4.10)

Next we set h = tg, for g ∈ 𝔤 and we differentiate (4.10) w.r.t. t. (Differentiating is allowed by (4.5)). We obtain (Ñ_{π}v|ϕ_{π}(g)u) = (ϕ_{π}(g)v|Ñ_{π}u) − i(v|ϕ_{π}(ig)u). (4.11) Substituting ig for g in (4.11) we obtain (Ñ_{π}v|ϕ_{π}(ig)u) = −(ϕ_{π}(ig)v|Ñ_{π}u) + i(v|ϕ_{π}(g)u). (4.12) Adding up (4.11) and (4.12) we get (Ñ_{π}v|a_{π}(g)u) = (a_{π}^{∗}(g)v|Ñ_{π}u) − (v|a_{π}(g)u), u, v ∈ D(Ñ_{π}). (4.13)

Next let us assume that u ∈ D(Ñπ^{\frac{3}{2}} ). Then Ñ_{π}u ∈ D(Ñπ^{\frac{1}{2}} ) ⊂ D(a_{π}(g)). Hence (4.13) implies (Ñ_{π}v|a_{π}(g)u) = (v|a_{π}(g)(Ñ_{π} − 1l)u). (4.14) Therefore, a_{π}(g)u ∈ D(Ñ_{π}) and we have Ñ_{π}a_{π}(g)u = a_{π}(g)(Ñ_{π} − 1)u, (4.15) or equivalently (Ñ_{π} + λ)a_{π}(g)u = a_{π}(g)(Ñ_{π} + λ − 1)u. (4.16) Now let v ∈ D(Ñπ^{\frac{1}{2}} ) and λ > 1. Then (Ñ_{π} + λ − 1)^{−1}v ∈ D(Ñπ^{\frac{3}{2}} ). Therefore, by (4.16) (Ñ_{π} + λ)a_{π}(g)(Ñ_{π} + λ − 1)^{−1}v = a_{π}(g)v. (4.17) Multiplying this with (Ñ_{π} + λ)^{−1} we obtain a_{π}(g)(Ñ_{π} + λ − 1)^{−1}v = (Ñ_{π} + λ)^{−1}a_{π}(g)v. (4.18)

Since linear combinations of functions (Ñ_{π} + λ)^{−1} with λ > 0 are strongly dense in the Von Neumann algebra of functions of Ñ_{π}, and a_{π}(g) is closed, (4.18) implies

a_{π}(g)F (Ñ_{π} − 1)v = F (Ñ_{π})a_{π}(g)v, v ∈ D(Ñπ^{\frac{1}{2}} )

for any bounded Borel function F. □

Lemma 4.5 K_{π} = {0} implies D(n_{π}) = {0}.

Proof. Suppose that D(n_{π}) ≠ {0}. We know that Ñ_{π} ≥ 0. Therefore, σ(Ñ_{π}) is nonempty and bounded from below. Hence λ_{0} := inf σ(Ñ_{π}) is a finite number, and Ran1l_{[λ_{0},λ_{0}+1[}(Ñ_{π}) ≠ {0}. By Lemma 4.4, for any h ∈ 𝔥

a_{π}(h)1l_{[λ_{0},λ_{0}+1[}(Ñ_{π}) = 1l_{[λ_{0}−1,λ_{0}[}(Ñ_{π})a_{π}(h). (4.19)

But

1l_{[λ_{0}−1,λ_{0}[}(Ñ_{π}) = 0.

Therefore, (4.19) is zero and

Ran1l_{[λ_{0}−1,λ_{0}[}(Ñ_{π}) ⊂ K_{π}.

□

The following lemma is immediate:

Lemma 4.6 Suppose that H = H_{0} ⊕ H_{1}. Suppose that

𝔥 ∋ h ↦ W_{π}(h) ∈ H

is a CCR representation and W_{π}(h) leave H_{0} invariant. Then W_{π}(h) leave also H_{1} invariant. Thus we have two CCR representations

𝔤 ∋ h ↦ W_{π}(h)|_{H_{0}},
𝔤 ∋ h ↦ W_{π}(h)|_{H_{1}}.

Let K_{π,i}, Ñ_{π,i} denote the corresponding spaces of vacua and the operators defined by (4.4) for the representations i = 0, 1. Then K_{π} = K_{π,0} ⊕ K_{π,1}, (4.20)

Ñ_{π} = Ñ_{π,0} ⊕ Ñ_{π,1}. (4.21)

Lemma 4.7 The operators W_{π}(h) preserve RanΩ_{π} for h ∈ 𝔤.

Proof. Since Ω_{π} is isometric, Ω_{π}(K_{π} ⊗ Γ_{fin}(𝔤)) is dense in RanΩ_{π}. It is also preserved by ϕ_{π}(h), h ∈ 𝔤, and consists of vectors analytic for ϕ_{π}(h). Hence, it is also preserved by W_{π}(h) = e^{iϕ_{π}(h)}. □

Proof of Theorem 4.3. By Lemma 4.7, we are in the situation of Lemma 4.6 and we have two CCR representations in H_{0} = RanΩ_{π} and in H_{1} = H_{0}^{⊥}.

By the definition of N_{π}, we have

N_{π} = N_{π,0} ⊕ N_{π,1},

where D(N_{π,1}) = {0}. We check immediately by (4.3) that Ñ_{π,0} = N_{π,0}.

We know that K_{π} ⊂ H_{0}, hence K_{π,1} = {0}. By Lemma 4.5, this implies D(Ñ_{1,π}) = {0}. Therefore, Ñ_{π} = N_{π}. □

5 Gaussian random processes and the Q-space representation

In this section we describe the Q−space representation of Fock space and discuss the notion of Wick ordering associated to a Q−space representation. We also recall the notion of hypercontractivity, following [S-H.K, Si1].

5.1 Gaussian processes

Let 𝔣 be a real Hilbert space with the scalar product (h_{1}, h_{2}), h_{1}, h_{2} ∈ 𝔣. Let Q be a space with a σ-algebra Q and a probability measure μ. Let Exp(F) denote ∫_{Q} F dμ, for any measurable function F on Q. A linear map 𝔣 ∋ h ↦ ϕ(h) (5.1)

into measurable functions on Q is called a Gaussian random process if

Exp (ϕ(h)^{2p}) = 2^{−p}(h, h)^{p}, Exp (ϕ(h)^{2p+1}) = 0,

or equivalently

Exp (e^{iϕ(h)}) = exp (−\frac{1}{2}(h, h)) .

Proposition 5.1 The following conditions are equivalent:

(1) Q is the smallest σ-algebra for which ϕ(h), h ∈ 𝔣 are measurable;

(2) L^{2}(Q, μ) is spanned by ϕ(h)^{n}, h ∈ 𝔣;

(3) L^{∞}(Q, μ) is the smallest W^{∗}-algebra containing e^{iϕ(h)}, h ∈ 𝔣.

We refer to [Si1, Lemma I.5] for the proof.

If the conditions of the above proposition are satisfied, then one says that the process (5.1) is full. If not, one can always make it full by choosing a smaller σ-algebra of subsets of Q. (Obviously, this procedure does not change the Hilbert space spanned by ϕ(h), nor the W^{∗}- algebra spanned by e^{iϕ(h)}). Let us assume that the random process (5.1) is full.

Let P_{n} denote the projection onto polynomials of degree n in ϕ(h), h ∈ 𝔣, inside L^{2}(Q, μ). For any h_{1}, …, h_{n} ∈ 𝔣 we define

:ϕ(h_{1}) ⋯ ϕ(h_{n}): := (1l − P_{n−1})ϕ(h_{1}) ⋯ ϕ(h_{n}).

We recall the well known Wick identities

\begin{matrix}:ϕ(h)^{n}:= ∑_{m=0}^{[n/2]} \frac{n!}{m!(n−2m)!}ϕ(h)^{n−2m}(−\frac{1}{2}(h, h))^{m}, \\ ϕ(h)^{n} = ∑_{m=0}^{[n/2]} \frac{n!}{m!(n−2m)!} :ϕ(h)^{n−2m}: (\frac{1}{2}(h, h))^{m}. \end{matrix} (5.2)

5.2 Q−space representation of Fock space

Let 𝔥 be a Hilbert space with a complex conjugation c, that is an antilinear map c : 𝔥 → 𝔥 such that c^{2} = 1 and (ch|cg) = (g|h). We set 𝔥_{c} := {h ∈ 𝔥 | ch = h}. Let 𝔐_{c} ⊂ B(Γ(𝔥)) be the abelian Von Neumann algebra generated by the Weyl operators W (h) for h ∈ 𝔥_{c}. The following basic result follows from the fact that Ω is a cyclic vector for 𝔐_{c} (see eg [S-H.K]).

Theorem 5.2 There exists a compact Hausdorff space Q, a probability measure μ on Q and a unitary map R such that

R : Γ(𝔥) → L^{2}(Q, dμ),
RΩ = 1,
R𝔐_{c}R^{∗} = L^{∞}(Q, dμ).

where 1 ∈ L^{2}(Q, dμ) is the constant function equal to 1 on Q. Moreover,

RΓ(c)u = \overline{Ru}, u ∈ Γ(𝔥),

and 𝔥_{c} ∋ h ↦ Rϕ(h)R^{∗} is a full Gaussian random process on Q.

The space L^{2}(Q, dμ) is called the Q−space representation of the Fock space Γ(𝔥) associated to 𝔐_{c}.

The following property of the Q−space representation is often useful (see [Si1, Prop.1.7]).

Proposition 5.3 Let 𝔥 = 𝔥_{1} ⊕ 𝔥_{2}, where 𝔥_{i}, i = 1, 2 are Hilbert spaces with conjugations c_{i}. Equip 𝔥 with the conjugation c = c_{1} ⊕ c_{2}. Then as Q−space representation of the Fock space Γ(𝔥) one can take L^{2}(Q, dμ) for

Q = Q_{1} × Q_{2}, μ = μ_{1} ⊗ μ_{2},

where L^{2}(Q_{i}, dμ_{i}), i = 1, 2 is the Q−space representation of Γ(𝔥_{i}). We have

RU = R_{1} ⊗ R_{2},

where U : Γ(𝔥_{1}) ⊗ Γ(𝔥_{2}) → Γ(𝔥) is defined in Subsect. 3.8.

To simplify notation, we will often omit the unitary transformation R in the formulas. Similarly a function V on Q will be identified with the operator of multiplication by V on Γ(𝔥) ≡ L^{2}(Q, dμ).

In particular an element of Γ(𝔥) ≡ L^{2}(Q, dμ) can be considered as a multiplication operator on Γ(𝔥), ie as an unbounded operator affiliated to the Von Neumann algebra 𝔐_{c}. For v ∈ Γ(𝔥) this operator will be denoted by Wick_{c}(v). It is the unique operator affiliated to 𝔐_{c} such that Wick_{c}(v)Ω = v. For instance, if h ∈ 𝔥 = ⊗_{s}^{1}𝔥:

Wick_{c}(h) = a^{∗}(h) + a(ch).

One can generalize this formula to an arbitrary v ∈ Γ_{fin}(𝔥), by writing Wick_{c}(v) as a Wick polynomial.

Proposition 5.4 Let v ∈ Γ_{fin}(𝔥). Then

Wick_{c}(v) = Wick(γ_{c}(v)),

where

γ_{c}(v) : Γ_{fin}(𝔥) ↦ Γ_{fin}(𝔥)

is defined by (u_{2}|γ_{c}(v)u_{1}) := (ĨΓ(c)u_{1} ⊗ u_{2}|N!^{− \frac{1}{2}} v), u_{1}, u_{2} ∈ Γ_{fin}(𝔥). (5.3)

Proof. The proposition follows from Prop. 5.6 below. In fact by linearity we may assume that v = Π_{1}^{p}a^{∗}(h_{i})Ω. Using then the concrete expression of γ_{c}(v) given in Prop. 5.6 i) we easily check the proposition. □

Proposition 5.5

i) γ_{c}(v)Ω = v,

ii) γ_{c}(v)Γ(c) = Γ(c)γ_{c}(Γ(c)v),

iii) if v ∈ ⊗_{s}^{p}𝔥, γ_{c}(v) ∈ ⊕_{r=0}^{p} B(⊗_{s}^{r}𝔥, ⊗_{s}^{p−r}𝔥).

Using the identity (3.16) (which is also true for antilinear q) we see that ii) in Prop. 5.5 is equivalent to the identity

Γ(c)Wick_{c}(v) = Wick_{c}(Γ(c)v)Γ(c),

which in turns follows from the fact that Γ(c) is simply the complex conjugation on L^{2}(Q, dμ).

Proposition 5.6 i) Let v = Π_{1}^{p}a^{∗}(h_{i})Ω. Then:

Wick_{c}(v) = ∑_{I⊂{1,...,p}} Π_{i∈I}a^{∗}(h_{i})Π_{i∉I}a(ch_{i}),

γ_{c}(v) = ∑_{I⊂{1,...,p}} | ⊗_{s}_{i∈I} h_{i})(⊗_{s}_{i∉I} ch_{i}|.

ii) (Wick’s theorem). For h ∈ 𝔥_{c}:

:ϕ(h)^{p}:= \frac{1}{2^{p/2}} ∑_{0}^{p} ( \begin{matrix}r \\ p \end{matrix} ) a^{∗}(h)^{r}a(h)^{p−r}.

Proof. it is easy to verify that the operator in the right hand side of the first identity of i) commutes with ϕ(h), h ∈ 𝔥_{c} and maps Ω onto v. Hence it equals Wick_{c}(v). The second identity of i) follows then from Thm. 3.12 v). To prove ii), we first claim that :ϕ(h)^{p}: Ω = \frac{1}{2^{p/2}} a^{∗}(h)^{p}Ω. In fact the r.h.s. is orthogonal to the polynomials in ϕ(h_{i}), h_{i} ∈ 𝔥_{c} of order less than p − 1 and differs from ϕ(h)^{p}Ω by a polynomial of order less than p − 1. Hence it equals :ϕ(h)^{p}: Ω. Now ii) follows from i) and the fact that since :ϕ(h)^{p}: is affiliated to 𝔐_{c}, Wick_{c}(:ϕ(h)^{p}:)Ω =:ϕ(h)^{p}:. □

If 𝔥 = L^{2}(IR, dk) and the conjugation c is defined by h(k) = h̄(−k), (which will be the case in the P (φ)_{2} theory), and v ∈ ⊗_{s}^{p}L^{2}(IR), then using the notation (3.13) we can write:

Wick_{c}(v)

= ∑_{r=0}^{p} ( \begin{matrix}p \\ r \end{matrix} ) p!^{− \frac{1}{2}} ∫ w(k_{1}, …, k_{r}, k_{r+1}, …, k_{p})

×a^{∗}(k_{1}) … a^{∗}(k_{r})a(−k_{r+1}) … a(−k_{p})dk_{1} ⋯ dk_{p}.

5.3 Hypercontractive semigroups

Let (Q, μ) be a measure probability space.

Definition 5.7 Let H_{0} ≥ 0 be a selfadjoint operator on H = L^{2}(Q, dμ). The semigroup e^{−tH_{0}} is hypercontractive if

i) e^{−tH_{0}} is a contraction on L^{1}(Q, dμ) for all t > 0,

ii) ∃ T, C such that

∥e^{−T H_{0}}ψ∥_{L^{4}(Q,dμ)} ≤ C∥ψ∥_{L^{2}(Q,dμ)}.

The abstract result used to construct the P (φ)_{2} Hamiltonian is the following theorem, due to Segal ([Se]).

Theorem 5.8 Let e^{−tH_{0}} be a hypercontractive semigroup. Let V be a real function on Q such that V ∈ L^{p}(Q, dμ) for some p > 2 and e^{−tV} ∈ L^{1}(Q, dμ) for all t > 0. Let V_{n} = 1l_{{|V |≤n}}V and H_{n} = H_{0} + V_{n}. Then the semigroups e^{−tH_{n}} converge strongly on H when n → ∞ to a strongly continuous semigroup on H denoted by e^{−tH}. Its infinitesimal generator H has the following properties:

i) H is the closure of H_{0} + V defined on D(H_{0}) ∩ D(V),

ii) H is bounded below:

H ≥ −c − ln ∥e^{−V} ∥_{L^{p}(Q,dμ)},

where c and p depend only on the constants C and T in Def. 5.7.

The following technical result (see [Si1, Lemma V.5] for a proof) will be used later to show that a given function V on Q verifies e^{−tV} ∈ L^{1}(Q, dμ).

Lemma 5.9 Let for κ ≥ 1, V_{κ}, V be functions on Q such that for some n ∈ IN

\begin{matrix}∥V − V_{κ}∥_{L^{p}(Q,dμ)} ≤ C(p − 1)^{n}κ^{−ϵ}, \\ V_{κ} ≥ −C(ln κ)^{n}. \end{matrix} (5.4)

Then

μ{q ∈ Q|V (q) ≤ −2C(ln κ)^{n}} ≤ Ce^{−cκ^{α}}

for some α > 0. Consequently e^{−tV} ∈ L^{1}(Q, dμ), ∀t > 0.

The following theorem of Nelson (see [Si1, Thm. 1.17]) establishes a connection between contractions on 𝔥 and hypercontractive semigroups on L^{2}(Q, dμ).

Theorem 5.10 Let r ∈ B(𝔥) be a selfadjoint contraction commuting with c. Then i) UΓ(r)U^{∗} is a positivity preserving contraction on L^{p}(Q, dμ), 1 ≤ p ≤ ∞. ii) if ∥r∥ ≤ (p−1)^{\frac{1}{2}} (q −1)^{− \frac{1}{2}} for 1 < p, q < ∞ then UΓ(r)U^{∗} is a contraction from L^{p}(Q, dμ)

to L^{q}(Q, dμ).

Combining Thm. 5.10 with Thm. 5.8, we obtain the following result.

Theorem 5.11 Let 𝔥 be a Hilbert space with a conjugation c. Let a be a selfadjoint operator on 𝔥 with [a, c] = 0, a ≥ m > 0. (5.5)

Let L^{2}(Q, dμ) be the Q−space representation of Γ(𝔥) and let V be a real function on Q with V ∈ L^{p}(Q, dμ) for some p > 2 and e^{−tV} ∈ L^{1}(Q, dμ) for all t > 0. Then:

i) the operator sum H = dΓ(a) + V is essentially selfadjoint on D(dΓ(a)) ∩ D(V).

ii) H ≥ −C, where C depends only on m and ∥e^{−V} ∥_{L^{p}(Q,dμ)}, for some p depending only on m.

Note that by applying Thm. 5.10 to a = (q − 1)^{− \frac{1}{2}} 1l_{𝔥} for q > 2, we obtain the following lemma about the L^{p} properties of finite vectors in Γ(𝔥) (see [Si1, Thm. 1.22]).

Lemma 5.12 Let ψ ∈ ⊗_{s}^{n}𝔥 and q ≥ 2. Then

∥Rψ∥_{L^{q}(Q,dμ)} ≤ (q − 1)^{n/2}∥ψ∥.

6 The spatially cut-off P (φ)_{2} Hamiltonian

In this section, we recall some standard facts about the construction of the spatially cut-off P (φ)_{2} Hamiltonian. Some of these facts are presented in a slightly more general form, which will be useful later when we will consider the Mourre theory for P (φ)_{2} Hamiltonians.

6.1 The spatially cut-off P (φ)_{2} model

We recall now the definition of the spatially cut-off P (φ)_{2} model that we will study in this paper (see for example [GJ1, S-H.K]). P (φ)_{2} models describe quantum field theories in 2 space-time dimensions, which means that the 1−particle Hilbert space 𝔥 is taken equal to L^{2}(IR, dk). We normalize the Fourier transform by

F : L^{2}(IR, dx) → L^{2}(IR, dk),
Fχ(k) = χ̂(k) = ∫ e^{−ik·x}χ(x)dx.

The complex conjugation is the map c defined by ch(k) = \overline{h}(−k), which corresponds to the usual conjugation f ↦ \overline{f} on L^{2}(IR, dx) by Fourier transformation. The bosonic Fock space Γ(𝔥) will be denoted by H.

We fix the dispersion relation

IR ∋ k ↦ ω(k) = (k^{2} + m^{2})^{\frac{1}{2}} , m > 0.

The kinetic energy is H_{0} = dΓ(ω).

Let us now define the interaction term. Let

φ(x) := ∫ e^{−ik·x} (a^{∗}(k) + a(−k)) \frac{dk}{ω(k)^{\frac{1}{2}}} (6.1)

be the local relativistic field operator defined in distribution sense.

Note that the local field φ(x) is denoted by a different variety of the letter phi than the Segal field ϕ(h), h ∈ 𝔥 (see Subsect. 2.2). It is useful to note that φ(x) can be formally expressed in terms of ϕ as follows. Set

f(k) := \sqrt{2}ω(k)^{− \frac{1}{2}} .

Let τ_{x} : L^{2}(IR, dk) → L^{2}(IR, dk) be the translation by x, that is τ_{x}h = e^{−ik·x}h. Then formally

φ(x) = ϕ(τ_{x}f). (6.2)

Unfortunately, τ_{x}f ∉ L^{2}(IR), so (6.2) has to be understood in distribution sense.

To remedy this one introduces UV-cutoff fields. Let χ ∈ L_{IR}^{1}(IR, dx) ∩ L^{2}(IR, dx) with ∫ χ(x)dx = 1 and let κ ≥ 1 be a large UV-cutoff parameter. We introduce for later use the cutoff fields

\begin{aligned}φ_{κ}(x) & := κ ∫ φ(y)χ(κ(y − x))dy \\ & = ∫ e^{−ik·x}χ̂(\frac{k}{κ}) (a^{∗}(k) + a(−k)) _{\frac{dk}{ω(k)^{\frac{1}{2}}}} . \end{aligned} (6.3)

If one sets

f_{κ}(k) := \sqrt{2}ω(k)^{− \frac{1}{2}} χ̂(\frac{k}{κ}), (6.4)

then one can write

φ_{κ}(x) = ϕ(τ_{x}f_{κ}).

Note that since the function τ_{x}f_{κ}(k) = e^{−ik·x}χ̂(\frac{k}{κ})ω(k)^{− \frac{1}{2}} belongs to 𝔥_{c}, φ_{κ}(x) is affiliated to the algebra 𝔐_{c}. We will set φ_{∞}(x) := φ(x).

To define the spatially cut-off P (φ)_{2} interaction, we fix a real polynomial of degree 2n:

P (λ) = ∑_{j=0}^{2n} a_{j}λ^{j}, with a_{2n} > 0, (6.5)

and a real function g ∈ L_{IR}^{1}(IR, dx) ∩ L^{2}(IR, dx) with g ≥ 0. We set for κ < ∞:

V_{κ} := ∫ g(x) :P (φ_{κ}(x)): dx (6.6)

which is an unbounded operator affiliated to 𝔐_{c}. We will see later that, when κ → ∞, V_{κ} converges in L^{2}(Q, dμ) to a function V, which we will denote by

V = ∫ g(x) :P (φ(x)): dx.

Alternatively one can view the multiplication operators V_{κ} and V as Wick polynomials, using the discussion in Subsect. 5.2. In fact for p ∈ IN, we have

\begin{matrix}∫ g(x) :φ_{κ}(x)^{p}: dx \\ = ∑_{r=0}^{p} ( \begin{matrix}p \\ r \end{matrix} ) ∫ w_{p,κ}(k_{1}, …, k_{r}, k_{r+1}, …, k_{p}) \\ ×a^{∗}(k_{1}) ⋯ a^{∗}(k_{r})a(−k_{r+1}) ⋯ a(−k_{p})dk_{1} ⋯ dk_{p}, \end{matrix} (6.7)

for

w_{p,κ}(k_{1}, ⋯, k_{p}) = ĝ(∑_{1}^{p} k_{i})Π_{1}^{p}χ̂(\frac{k_{i}}{κ})ω(k_{i})^{− \frac{1}{2}} . (6.8)

From Lemma 6.1 and Prop. 6.5 below, we deduce that V and V_{κ} are defined as unbounded operators with domain D(N)^{n}. This implies that the operator sum H_{0} + V is well defined as a symmetric operator on D(H_{0}) ∩ D(N^{n}).

The construction of a unique selfadjoint extension of H_{0} + V is outlined in Subsect. 6.4. This Hamiltonian is denoted by

H = H_{0} + ∫ g(x) :P (φ(x)): dx

and is called a spatially cut-off P (φ)_{2} Hamiltonian or simply a P (φ)_{2} Hamiltonian.

6.2 Assumptions on g

In this subsection, we discuss the various assumptions on the cutoff function g which will be used in our paper.

A spatially cut-off P (φ)_{2} model is completely specified by the polynomial P and the cutoff g. Let us introduce the following assumptions:

(A) g ≥ 0, g ∈ L_{IR}^{1}(IR, dx) ∩ L^{2}(IR, dx).

Assumption (A) is a standard assumption needed to construct H as a selfadjoint operator. (This assumption can be relaxed to g ≥ 0, g ∈ L_{IR}^{1}(IR, dx) ∩ L^{1+ϵ}(IR, dx) ϵ > 0, see [Si1]).

(C) g ∈ H^{\frac{1}{2}} (IR).

Assumption (C) will be needed if degP = 4 to ensure that D(H) = D(H_{0}) ∩ D(V). This fact allows to give a simpler treatment of the Mourre theory for φ_{2}^{4} Hamiltonians that does not use the Q−representation.

(Mm) (x · ∂_{x})^{j}g ∈ L^{2}(IR, dx), j = 1, …, m.

Assumption (Mm) is needed to define the commutators ad_{dΓ(a)}^{m}V where a is the generator of dilations on 𝔥 as densely defined operators (a priori they are only defined as quadratic forms).

(Is) ⟨x⟩^{s}g ∈ L^{2}(IR, dx), s ≥ 0.

Assumption (Is) is needed for the scattering theory of spatially cut-off P (φ)_{2} Hamiltonians. In particular (Is) for s > 1 is a short-range condition, under which the asymptotic fields and the wave operators can be constructed.

(Bm) g(x) ≤ Cg(y)⟨x − y⟩^{N}, |(x · ∂_{x})^{j}g(x)| ≤ Cg(x), 0 ≤ j ≤ m.

Assumption (Bm) will be needed in Section 8 in order to be able to control the commutator ad_{dΓ(a)}^{m}V. Note that (Bm) implies (Mm).

We will always assume (A). All other assumptions will be explicitly stated.

6.3 Some properties of the interaction kernel

We collect here various properties of the interaction kernels w_{p,κ} of the Wick polynomials V_{κ}. The following lemma is well known.

Lemma 6.1 The kernels w_{p,κ} are in L^{2}(IR^{p}) for 1 ≤ κ ≤ ∞ and ∥w_{p,κ} − w_{p,∞}∥_{L^{2}(IR^{p})} ≤ C∥g∥_{L^{2}(IR)}κ^{−ϵ}, ϵ > 0.

Proof. We use the bound

Π_{1}^{p}a_{j} ≤ ∑_{i=1}^{p}(Π_{j≠i}a_{j})^{p/(p−1)}, (6.9)

which follows from the fact that

(Π_{1}^{p}λ_{i})^{1/p} ≤ ∑_{1}^{p} λ_{i},

applied to λ_{i} = Π_{j≠i}a_{j}^{p/(p−1)}. Applying (6.9) to a_{i} = ω(k_{i})^{− \frac{1}{2}} we obtain that w_{p,∞} and hence w_{p,κ} for κ < ∞ belong to L^{2}(IR^{d}). The bound on ∥w_{p,κ} − w_{p,∞}∥ is a direct computation, using (6.9). □

We deduce from Lemma 6.1 the following result:

Lemma 6.2 The operators V_{κ}(N + 1)^{−n} are bounded on H for 1 ≤ κ ≤ ∞ and ∥(V − V_{κ})(N + 1)^{−n}∥ ≤ C∥g∥_{L^{2}(IR)}κ^{−ϵ}, ϵ > 0.

Lemma 6.3 Let j ∈ C^{∞}(IR) with j ≡ 0 near 0 and j ≡ 1 near infinity. Then for 1 ≤ i ≤ p:

∥j(\frac{x_{i}}{R})w_{p}∥_{L^{2}(IR^{p})} ∈ \{ \begin{matrix}O(R^{−s}) under hypothesis (Is), \\ o(R^{0}) under hypothesis (A). \end{matrix}

Proof. It suffices to prove the lemma for i = 1.

It follows from (Is) that ĝ belongs to the Sobolev space H^{s}(IR). Let us first check that

|D_{k_{1}}|^{s}w_{p} ∈ L^{2}(IR^{p}). (6.10)

By interpolation it suffices to check this for s ∈ IN. We see that ∂_{k_{1}}^{s}w_{p} is a sum of terms of the form

ĝ^{(s_{1})}(∑_{1}^{p} k_{i})ω^{− \frac{1}{2}} (k_{1})^{(s_{2})}Π_{2}^{p}ω(k_{i})^{− \frac{1}{2}} ,

where s = s_{1} + s_{2} and hence ĝ^{(s_{1})} ∈ L^{2}(IR). Since ĝ^{(s_{1})} belongs to L^{2}(IR, dk), the bound (6.9) gives that ∂_{k_{1}}^{s}w_{p} ∈ L^{2}(IR^{d}), and hence (6.10) is true. Next we note that since j vanishes near the origin, j(x) = |x|^{s}j_{s}(x), where j_{s} is bounded. Now

j(\frac{x_{1}}{R}) = j(\frac{−D_{k_{1}}}{R}) = R^{−s}j_{s}(\frac{−D_{k_{1}}}{R})|D_{k_{1}}|^{s}.

and by the spectral theorem, j_{s}(\frac{−D_{k_{1}}}{R}) is a uniformly bounded operator on L^{2}(IR^{p}). The lemma follows then from (6.10). □

6.4 Existence and basic properties

We summarize now standard results on the existence of the P (φ)_{2} Hamiltonian( see [GJ1, Se, S-H.K, Ro1]).

Theorem 6.4 Let P be a real, bounded below polynomial of degree 2n and g ≥ 0, g ∈ L_{IR}^{1}(IR, dx) ∩ L^{2}(IR, dx). Then:

i) the Hamiltonian H = H_{0} + V for V = ∫ g(x) : P (φ(x)) : dx is essentially selfadjoint on D(H_{0}) ∩ D(V). ii) there exist b, C ≥ 0 such that the following first order estimates hold

H_{0} ≤ C(H + b), (6.11)

N ≤ C(H + b). (6.12)

Proof. i) follows from Thm. 5.11 using Lemma 6.6 below. ii) follows from Thm. 5.11 with a replaced by (1 − ϵ)a. (6.12) follows from (6.11). □

Proposition 6.5 Let P (λ) be a real polynomial of degree 2n as in (6.5) and P̃(λ) be a real polynomial of degree ≤ 2n. Assume that the coefficient ã_{2n} of P̃(λ) and a_{2n} of P (λ) satisfy |ã_{2n}| < a_{2n}. Let g, g̃ ∈ L_{IR}^{1}(IR, dx) ∩ L^{2}(IR, dx) be two functions with g ≥ 0, |g̃| ≤ g. Let

Ṽ := ∫ g̃(x)P̃(φ(x)): dx.

Then Ṽ is a multiplication operator in the Q−space representation and there exist C, b such that

|Ṽ | ≤ C(H + b).

The above theorem and proposition follow from the next lemma, where we collect some well-known properties (see eg [Ne, Se, S-H.K]) which show that the P (φ)_{2} interaction is a multiplication operator in the Q−space representation.

Lemma 6.6 Under the conditions of Proposition 6.5, set

W := C ∫ g(x) :P (φ(x)): −g̃(x) :P̃(φ(x)): dx.

Then W is a multiplication operator in the Q−space representation. Moreover W ∈ L^{p}(Q, dμ) for all p < ∞ and ∥e^{−tW} ∥_{L^{1}(Q,dμ)} depends only on P, P̃ and ∥g∥_{L^{1}(IR)∩L^{2}(IR)}.

Proof. Note first that since |g̃| ≤ g, ∥g̃∥_{L^{1}(IR)∩L^{2}(IR)} is less than ∥g∥_{L^{1}(IR)∩L^{2}(IR)}. Let for κ ≥ 1 W_{κ} be the cutoff operator defined as in (6.6). Both W and W_{κ} are Wick polynomials. Applying Lemma 6.2 we see that W (N + 1)^{−n} is bounded and

∥(W − W_{κ})(N + 1)^{−n}∥ ≤ C∥g∥_{L^{2}(IR)}κ^{−ϵ}, ϵ > 0. (6.13)

We have seen at the end of Subsect. 6.1 that W_{κ} is a multiplication operator by a function W_{κ} on Q. Moreover W_{κ} ∈ L^{2}(Q, dμ) since ∥W_{κ}∥_{L^{2}(Q,dμ)} = ∥W_{κ}Ω∥_{H} < ∞. Next it follows from (6.13) that W_{κ} is Cauchy in L^{2}(Q, dμ) and hence converges to W in L^{2}(Q, dμ) when κ → +∞. We note then that W Ω ∈ ⊗_{s}^{2n}𝔥 since W is a Wick polynomial of degree 2n, which by Lemma 5.12 implies that W ∈ L^{p}(Q, dμ) for all p < ∞.

To bound ∥e^{−tW} ∥_{L^{1}(Q,dμ)}, we will use Lemma 5.9, checking that the constants C, ϵ, n there depend only on P, P̃ and ∥g∥_{L^{1}(IR)∩L^{2}(IR)}. The first bound of (5.4) follows from (6.13) and Lemma 5.12. To check the second bound, we use the Wick identities (5.2), which yield

\begin{matrix}:P (φ_{κ}(x)):= ∑_{p=0}^{2n} a_{p} ∑_{r=0}^{[p/2]} b_{p,r}∥φ_{κ}(x)Ω∥^{2r}φ_{κ}(x)^{p−2r}, \\ :P̃(φ_{κ}(x)):= ∑_{p=0}^{2n} ã_{p} ∑_{r=0}^{[p/2]} b_{p,r}∥φ_{κ}(x)Ω∥^{2r}φ_{κ}(x)^{p−2r}, \end{matrix} (6.14)

for b_{p,0} = 1. Hence as an inequality between functions on Q we have:

:P (φ_{κ}(x)): −| :P̃(φ_{κ}(x)): | ≥ F_{κ}(φ_{κ}(x)),

for

F_{κ}(λ) = ∑_{p=0}^{2n} ∑_{r=0}^{[p/2]} c_{p,r}∥φ_{κ}(x)Ω∥^{2r}λ^{p−2r}, (6.15)

and c_{2n,0} = a_{2n} − |ã_{2n}|. If we apply the bound a^{2r}b^{p−2r} ≤ ϵb^{p} + C_{ϵ}a^{p} to all terms in (6.15) for p < 2n and use that a_{2n} − |ã_{2n}| > 0, we obtain

F_{κ}(λ) ≥ −C(∥φ_{κ}(x)Ω∥^{2n} + 1). (6.16)

By a direct computation, we check that ∥φ_{κ}(x)Ω∥ = ∥φ_{κ}(0)Ω∥ ∈ O(ln κ^{\frac{1}{2}} ). This gives

W_{κ} = ∫ g(x) :P (φ_{κ}(x)): −g̃(x) :P̃(φ_{κ}(x)): dx
≥ ∫ g(x) :P (φ_{κ}(x)): −g(x)| :P̃(φ_{κ}(x)): |dx
≥ −C∥g∥_{L^{1}(IR)}((ln κ)^{n} + 1),

which proves the second bound of (5.4), with a constant depending only on ∥g∥_{L^{1}(IR)}. □

7 Higher order estimates

In this section we will state some higher order estimates which will be very important in the sequel. These higher order estimates are due to Glimm and Jaffe [GJ2] for the φ_{2}^{4} model and (in a more general form than here) to Rosen [Ro2] for the general P (φ)_{2} model. Note however that the proof in [Ro2] is only valid under the additional hypothesis that g ∈ C_{0}^{∞}(IR) (see Remark 7.5). In this subsection we will explain the modifications to Rosen’s proof necessary to treat the general case when g ∈ L_{IR}^{1}(IR, dx) ∩ L^{2}(IR, dx). The reader may also consult [Si2] for a review of higher order estimates.

Theorem 7.1 Assume hypothesis (A). Then there exists b > 0 such that for all α ∈ IN, the following higher order estimates hold:

\begin{matrix}∥N^{α}(H + b)^{−α}∥ < ∞, \\ ∥H_{0}N^{α}(H + b)^{−n−α}∥ < ∞, \\ ∥N^{α}(H + b)^{−1}(N + 1)^{1−α}∥ < ∞. \end{matrix} (7.1)

In the case of the φ_{2}^{4} model a better estimate is known.

Theorem 7.2 Assume degP = 4 and hypothesis (C). Then there exists b > 0 such that

∥H_{0}(H + b)^{−1}∥ < ∞.

Consequently D(H) = D(H_{0}) ∩ D(V).

Proof. The proof given in [GJ2] for g ∈ C_{0}^{∞}(IR) is still valid under hypothesis (C). In fact one first proves that if W = ∫ g(x) :φ_{κ}(x)^{p}: dx for p ≤ 4 then (see [GJ2, Equ. 2.17]):

∥(H_{0} + 1)^{−1}[H0^{\frac{1}{2}} , [H0^{\frac{1}{2}} , W]](H_{0} + 1)^{−1}∥ ≤ C∥ω^{\frac{1}{2}} (∑_{1}^{4} k_{i})w_{p}∥_{L^{2}(IR^{4})}.

(Note that the expressions W, [H0^{\frac{1}{2}} , W] and [H0^{\frac{1}{2}} , [H0^{\frac{1}{2}} , W]] are well defined as quadratic forms

on S = Γ_{fin}(C_{0}^{∞}(IR)) since H0^{\frac{1}{2}} preserves S). One uses then Jaffe’s double commutator trick (see eg [Si2, Sect. 4]) to obtain that as quadratic forms on S

H^{2} ≥ c(H_{0}^{2} + V^{2}) − d. (7.2)

Now it follows from the higher order estimates that any core for H_{0}^{n} is a core for H, and in particular S is a core for H. Hence (7.2) extends to D(H), which proves the theorem. □

Remark 7.3 The importance of the higher order estimates comes from the fact that the domain of H is not known explicitly. In particular, the question if D(H) = D(H_{0})∩D(V) is still an open problem, except for the φ_{2}^{4} model, where this result was shown by Glimm and Jaffe in [GJ2]. This means that for u ∈ D(H) the identity Hu = H_{0}u+V u does not make sense for the general P (φ)_{2} model. However a consequence of the higher order estimates is that D(H^{n}) ⊂ D(V) ∩ D(H_{0}) so that this identity makes sense for u ∈ D(H^{n}).

The proof of higher order estimates in [Ro2] is based on the pullthrough formula, which gives an expression for the multi-commutators of annihilation operators a(k_{i}) with the resolvent (H − z)^{−1}. The technical problem is that in order to make sense of these formal computations one needs a subspace D of H which is in the domain of all powers of creation operators and on which H is essentially selfadjoint.

To circumvent this difficulty, Rosen introduced cutoff Hamiltonians for which the interaction acts only on a finite number of degrees of freedom. In a Q−space representation these cutoff Hamiltonians become differential operators, for which the construction of a subspace D with the above properties is easy. The higher order estimates are then shown for the cutoff Hamiltonians, with constants uniform in the cutoff parameters. The proof is then completed by taking the cutoff to infinity.

7.1 Cutoff Hamiltonians

In this subsection we introduce the U.V. cutoff Hamiltonians used in the proof of the higher order estimates.

Let 𝔥 be a Hilbert space equipped with a conjugation c. Let π_{1} : 𝔥 → 𝔥_{1} be an orthogonal projection on a closed subspace 𝔥_{1} of 𝔥 with [π_{1}, c] = 0. Let 𝔥_{1}^{⊥} be the orthogonal complement of 𝔥_{1}. In all formulas below we will consider π_{1} as an element of B(𝔥, 𝔥_{1}). With this convention the orthogonal projection on 𝔥_{1}, considered as an element of B(𝔥, 𝔥), is equal to π_{1}^{∗}π_{1}.

Let U : Γ(𝔥_{1}) ⊗ Γ(𝔥_{1}^{⊥}) → Γ(𝔥) the unitary map defined in Subsect. 3.8. We denote by L^{2}(Q_{1}, dμ_{1}), L^{2}(Q_{1}^{⊥}, dμ_{1}^{⊥}) the Q−space representations of Γ(h_{1}), Γ(𝔥_{1}^{⊥}). Recall that by Prop. 5.3, we may take as Q−space representation of Γ(𝔥) the space L^{2}(Q, dμ) for Q = Q_{1} × Q_{1}^{⊥}, μ = μ_{1} ⊗ μ_{1}^{⊥}. Accordingly we denote by (q_{1}, q_{1}^{⊥}) the elements of Q = Q_{1} × Q_{1}^{⊥}.

If W ∈ B(Γ(𝔥)) we set:

B(Γ(𝔥)) ∋ Π_{1}W := U(Γ(π_{1})W Γ(π_{1}^{∗}) ⊗ 1l)U^{∗}.

Lemma 7.4 i) If w ∈ B_{fin}(Γ(𝔥)) then

(7.3) Π_{1}Wick(w) = Wick(Γ(π_{1}^{∗}π_{1})wΓ(π_{1}^{∗}π_{1})).

ii) If V is a multiplication operator by a function in L^{2}(Q, dμ) then Π_{1}V is the operator of multiplication by the function

Π_{1}V (q_{1}) = ∫_{Q_{1}^{⊥}} V (q_{1}, q_{1}^{⊥})dμ_{1}^{⊥}. (7.4)

Proof. To prove i) we may assume by linearity that

w = | ⊗_{s}_{1}^{q} h_{i})(⊗_{s}_{1}^{p} g_{i}|,

for which the verification of i) is easy. To prove ii) we first deduce from Prop. 3.5 that Γ(π_{1})U = 1l ⊗ |Ω)(Ω|. This implies that as a multiplication operator on Q_{1}, Γ(π_{1})V Γ(π_{1}^{∗}) is given by (7.4). Then one uses Prop. 5.3. □

In particular if W = Π_{1}^{q}a^{∗}(h_{i})Π_{1}^{p}a(g_{i}), then

Π_{1}W = Π_{1}^{q}a^{∗}(π_{1}^{∗}π_{1}h_{i})Π_{1}^{p}a(π_{1}^{∗}π_{1}g_{i}). (7.5)

Let now {π_{n}}_{n∈IN} be a sequence of orthogonal projections on 𝔥 such that

π_{n} ≤ π_{n+1}, [π_{n}, c] = 0, s- lim_{n→+∞} π_{n} = 1l, (7.6)

and let Π_{n} the associated maps defined by (7.3). Using the representation (7.4) it is shown in [S-H.K, Prop. 4.9] that

i) Π_{n}V → V in L^{p}(Q, dμ), when n → ∞, if V ∈ L^{p}(Q, dμ),
ii) ∥e^{−tΠ_{n}V} ∥_{L^{1}(Q,dμ)} ≤ ∥e^{−tV} ∥_{L^{1}(Q,dμ)}.

(7.7)

Let us now specify the particular sequence of projections corresponding to the lattice regularization and volume cutoff used in [Ro2]. For v ≥ 1 we introduce the lattice v^{−1}ZZ and let

IR ∋ k ↦ [k]_{v} ∈ v^{−1}ZZ

be the integer part of k defined by −(2v)^{−1} < k − [k]_{v} ≤ (2v)^{−1}. For γ ∈ v^{−1}ZZ, we denote by e_{γ} ∈ 𝔥 the vector e_{γ}(k) = v^{\frac{1}{2}} 1l_{]−(2v)^{−1},(2v)^{−1}]}(k − γ) and set a^{♯}(γ) := a^{♯}(e_{γ}). For κ ∈ [1, +∞[ an UV cutoff parameter, we denote by Γ_{κ,v} the set v^{−1}ZZ ∩ {|γ| ≤ κ}, by 𝔥_{κ,v} the finite dimensional space spanned by {e_{γ}}_{γ∈Γ_{κ,v}}, and by π_{κ,v} the orthogonal projection from 𝔥 onto 𝔥_{κ,v}.

Let us fix a sequence (κ_{n}, v_{n}) tending to (∞, ∞) in such a way that

Γ_{κ_{n},v_{n}} ⊂ Γ_{κ_{n+1},v_{n+1}}, (7.8)

We denote by Γ_{n} the finite lattice Γ_{κ_{n},v_{n}} and choose as π_{n} the projection on 𝔥_{n} := 𝔥_{κ_{n},v_{n}}, which satisfies (7.6).

If V = ∫ g(x) :P (φ(x)): dx, we define the cutoff interaction:

V_{n} := Π_{n}V.

Using (7.5) and (6.7), we obtain the following explicit expression for V_{n}:

V_{n} = ∑_{p=0}^{degP} a_{p} ∫ g(x) :φ_{n}(x)^{p}: dx, (7.9)

where

:φ_{n}(x)^{p}:= ∑_{r=0}^{p} ( \begin{matrix}p \\ r \end{matrix} ) ∑_{γ_{1},...,γ_{p}∈Γ_{n}} a^{∗}(γ_{1}) ⋯ a^{∗}(γ_{r})a(−γ_{r+1}) ⋯ a(−γ_{p})Π_{1}^{p}μ_{n}(x, γ_{i}), (7.10)

and

μ_{n}(x, γ) := vn^{\frac{1}{2}} ∫_{−(2v_{n})^{−1}}^{+(2v_{n})^{−1}} e^{−i⟨x,γ+k⟩}ω(γ + k)^{− \frac{1}{2}} dk. (7.11)

Remark 7.5 Our definition of the cutoff interaction is different from the one used in [Ro2]. In fact the cutoff interaction used there is obtained by replacing the orthogonal projection π_{n} : 𝔥 → 𝔥_{n} by the unbounded operator:

h ↦ ∑_{γ∈Γ_{κn,vn}} v^{− \frac{1}{2}} h(γ)e_{γ}.

With this convention, it is easy to see that for example V_{n}Ω will not converge to V Ω for an arbitrary g ∈ L_{IR}^{1}(IR, dx) ∩ L^{2}(IR, dx).

To define the cutoff kinetic energy, we set as in [Ro2]:

\begin{matrix}ω_{n} : IR → IR, \\ ω_{n}(k) := ω([k]_{v_{n}}). \end{matrix}

Since [ω_{n}, π_{n}] = 0, the operator ω_{n} acts on 𝔥_{n} and 𝔥_{n}^{⊥}. By Prop. 3.5, we have:

(7.12) H_{0,n} := dΓ(ω_{n}) = U_{n} (dΓ(ω_{n}|_{𝔥_{n}}) ⊗ 1l + 1l ⊗ dΓ(ω_{n}|_{𝔥_{n}^{⊥}})) U_{n}^{∗},

where U_{n} is the unitary operator between Γ(𝔥_{n}) ⊗ Γ(𝔥_{n}^{⊥}) and Γ(𝔥). The cutoff Hamiltonian is then defined as:

H_{n} := H_{0,n} + V_{n}.

7.2 Properties of the cutoff Hamiltonians

In this subsection we collect some properties of the cutoff Hamiltonians which are needed to prove the higher order estimates. These properties are: existence of a suitable domain of essential selfadjointness, uniform lower bounds and finally resolvent convergence to the Hamiltonian H.

Proposition 7.6 (Ro2) The Hamiltonians H_{n}^{j}, j ∈ IN are essentially selfadjoint on

D_{n} = U(Γ_{fin}(𝔥_{n}) ⊗ Γ_{fin}(𝔥_{n}^{⊥} ∩ S(IR))).

Proof. As in [Ro2], we have:

U_{n}^{∗}H_{n}U_{n} = Ĥ_{n} ⊗ 1l + 1l ⊗ dΓ(ω_{n}|_{𝔥_{n}^{⊥}}),

for Ĥ_{n} = dΓ(ω_{n}|_{𝔥_{n}}) + V_{n}. Since 𝔥_{n} is finite dimensional, in the Q−space representation Ĥ_{n} becomes a differential operator −Δ + W (x) for W a bounded below polynomial, acting on L^{2}(IR^{dim𝔥_{n}}, dx). By [GJ3, Thm. 2.2.6] Ĥ_{n}^{j} is essentially selfadjoint on Γ_{fin}(𝔥_{n}) for j ∈ IN. The arguments in [Ro2] give then the proposition. □

Proposition 7.7 Let for n ∈ IN, g̃_{n} ∈ L_{IR}^{1}(IR, dx) ∩ L^{2}(IR, dx) with |g̃_{n}| ≤ Cg and P̃(λ) a real polynomial of degree less than degP − 1. Then there exist constants a, b > 0 such that

\begin{matrix}H_{0,n} ≤ a(H_{n} + b), \\ ∫ g̃_{n}(x) :P̃(φ_{n}(x)): dx ≤ C(H_{n} + b). \end{matrix}

Proof. Let

W^{n} = ∫ g(x) :P (φ(x)): −g̃_{n}(x) :P̃(φ(x)): dx,

and W_{n} = Π_{n}W^{n}. By Lemma 6.6 ∥e^{−tW^{n}}∥_{L^{1}(Q,dμ)} is bounded uniformly in n. Hence by (7.7) ∥e^{−tW_{n}}∥_{L^{1}(Q,dμ)} is bounded uniformly in n. On the other hand H_{0,n} = dΓ(ω_{n}), where ω_{n} satisfies (5.5). It follows then from Thm. 5.11 that H_{0,n} + W_{n} is bounded below uniformly in n. This shows the second bound in the proposition. The first one follows from the same argument, replacing H_{0,n} by (a − 1)H_{0,n}. □

Finally the following result is shown in [S-H.K, Prop. 4.8].

Proposition 7.8 For z ≤ −b:

(H_{n} − z)^{−1} → (H − z)^{−1} in norm

when n → +∞.

7.3 Proof of the higher order estimates

We now explain the modifications to the proof of Rosen [Ro2] needed in our case. The only places where modifications are needed are the ones where the interaction V_{n} appears, ie in [Ro2, Lemma 4.4]. We define for I = {1, …, s}, k_{i} ∈ IR, i ∈ I:

V_{n}^{I} := ad_{a(k_{1})} ⋯ ad_{a(k_{s})}V_{n},

which is well defined as an operator on D_{n}. The analog of [Ro2, Lemma 4.4] is now:

Lemma 7.9 There exist b, c > 0 such that for all λ_{1}, λ_{2} < −b, (H_{n} − λ_{2})^{− \frac{1}{2}} V_{n}^{I}(H_{n} − λ_{1})^{− \frac{1}{2}} defined on (H_{n} − λ_{1})^{\frac{1}{2}} D_{n} extends to a bounded operator on H such that

∥(H_{n} − λ_{2})^{− \frac{1}{2}} V_{n}^{I}(H_{n} − λ_{1})^{− \frac{1}{2}} ∥ ≤ cΠ_{i}^{s}ω(k_{i})^{− \frac{1}{2}} .

Proof. Using (7.10) and the commutation relation

[a(k), a^{∗}(γ)] = e_{γ}(k) = v^{\frac{1}{2}} δ(γ, [k]_{v}),

for

δ(γ, γ′) = \{ \begin{matrix}1 if γ = γ′, \\ 0 otherwise \end{matrix} ,

we obtain that:

[a(k),:φ_{n}^{p}(x):] = 0, if |k| > κ_{n},

and

\begin{aligned}[a(k),:φ_{n}^{p}(x):] & = ∑_{r=0}^{p} r ( \begin{matrix}p \\ r \end{matrix} ) ∑_{γ_{2},...,γ_{p}∈Γ_{n}} a^{∗}(γ_{2}) ⋯ a^{∗}(γ_{r})a(−γ_{r+1}) ⋯ a(−γ_{p}) \\ & ×vn^{\frac{1}{2}} μ_{n}(x, [k]_{v_{n}})Π_{2}^{p}μ_{n}(x, γ_{i}), \end{aligned}

if |k| ≤ κ_{n}.

We obtain

V_{n}^{I} = Π_{1}^{s}1l_{{|k|≤κ}}(k_{i})v_{n}^{s/2} ∫ g(x)Π_{1}^{s}μ_{n}(x, [k_{i}]_{v_{n}}) :P^{(s)}(φ_{n}(x)): dx.

For fixed (k_{1}, …, k_{s}), the operator V_{n}^{I} is of the form:

∫ g̃_{n}(x, k_{1}, …, k_{s}) :Q(φ_{n}(x)): dx

for

g̃_{n}(x, k_{1}, …, k_{s}) = v_{n}^{s/2}g(x)Π_{1}^{s}1l_{{|k|≤κ}}(k_{i})μ_{n}(x, k_{i}),

and Q(λ) = P^{(s)}(λ). Since

sup_{γ∈Γ_{v},|k|≤1} \frac{ω(γ)}{ω(γ + k)} ≤ c_{0},

we have:

|μ_{n}(x, k)| ≤ c_{0}ω(k)^{− \frac{1}{2}} v_{n}^{− \frac{1}{2}} ,

which yields

|g̃_{n}(x, k_{1}, …, k_{s})| ≤ c_{0}g(x)Π_{1}^{s}ω(k_{i})^{− \frac{1}{2}} .

Applying then Prop. 7.7, we obtain that

∥(H_{n} + b)^{− \frac{1}{2}} V_{n}^{I}(H_{n} + b)^{− \frac{1}{2}} ∥ ≤ CΠ_{1}^{s}ω(k_{i})^{− \frac{1}{2}} , uniformly in n,

which implies the lemma. □

7.4 Number energy estimates

In this subsection we state a consequence of the higher order estimates which will be used in the sequel. We will denote by H^{ext} the Hamiltonian H ⊗ 1l + 1l ⊗ H_{0}, acting on the extended Hilbert space H^{ext} = Γ(𝔥) ⊗ Γ(𝔥).

We will use the following notation: let an operator B(t) depending on some parameter t map ∩_{n}D(N^{n}) ⊂ H into itself. We will write

B(t) ∈ (N + 1)^{m}O_{N}(t^{p}) for m ∈ IR if (7.13)

∥(N + 1)^{−m−k}B(t)(N + 1)^{k}∥ ≤ C_{k}⟨t⟩^{p}, k ∈ ZZ.

If (7.13) holds for any m ∈ IR, then we will write

B(t) ∈ (N + 1)^{−∞}O_{N}(t^{p}).

Likewise, for an operator C(t) that maps ∩_{n}D(N^{n}) ⊂ H into ∩_{n}D((N_{0} + N_{∞})^{n}) ⊂ H^{ext} we will write C(t) ∈ (N + 1)^{m}Ǒ_{N}(t^{p}) for m ∈ IR if (7.14)

∥(N_{0} + N_{∞})^{−m−k}C(t)(N + 1)^{k}∥ ≤ C_{k}⟨t⟩^{p}, k ∈ ZZ.

If (7.14) holds for any m ∈ IR, then we will write

B(t) ∈ (N + 1)^{−∞}Ǒ_{N}(t^{p}).

The notation (N + 1)o_{N}(t^{p}), (N + 1)^{m}ǒ_{N}(t^{p}) are defined similarly.

Proposition 7.10 The following properties hold:

i) uniformly for z in a compact set of C we have:

(H − z)^{−1} ∈ (N + 1)^{−1}O_{N}(|Imz|^{−1}), m ∈ IR.

ii) for χ ∈ C_{0}^{∞}(IR) we have:

∥N^{m}χ(H)N^{p}∥ < ∞, m, p ∈ IN.

Proof. ii) follows directly from (7.1). To prove i), it is enough to prove that for m ∈ IN

(N + 1)^{m}(H − z)^{−1}(N + 1)^{1−m} ∈ O(|Imz|^{−1}), m ∈ IR.

We use an induction on m. For m = 0, i) follows from (7.1). Assume that i) holds for m − 1. Then we write:

N^{m}(H − z)^{−1}(N + 1)^{1−m}
= N^{m}(H + b)^{−1}(N + 1)^{1−m}(N + 1)^{m−1}(H + b)(H − z)^{−1}(N + 1)^{1−m}
= N^{m}(H + b)^{−1}(N + 1)^{1−m}(N + 1)^{m−1}(1l + (b + z)(H − z)^{−1})(N + 1)^{1−m}.

So i) for m follows from (7.1) and the induction hypothesis.□

7.5 Commutator estimates

In this subsection we estimate commutators between operators Γ(q), I(j) and the Hamiltonians H and H^{ext}. These estimates rely on the identities of Subsect. 3.12 and the higher order estimates.

The following lemma is analogous to [DG1, Lemma 3.3].

Lemma 7.11 Let q ∈ C_{0}^{∞}(IR^{d}), 0 ≤ q ≤ 1, q = 1 near 0. Set for R ≥ 1, where q^{R}(x) = q(\frac{x}{R}). Then for χ ∈ C_{0}^{∞}(IR):

[Γ(q^{R}), χ(H)] ∈ \{ \begin{matrix}(N + 1)^{−∞}O_{N}(R^{− inf(s,1)}) under hypothesis (Is), \\ (N + 1)^{−∞}o_{N}(R^{0}) under hypothesis (A). \end{matrix}

Proof. Let us prove the lemma under hypothesis (Is), the proof under hypothesis (A) being similar. We have [Γ(q^{R}), N] = 0, hence Γ(q^{R}) preserves D(N^{n}). By Lemma 3.4 ii)

[H_{0}, Γ(q^{R})] = dΓ(q^{R}, [ω, q^{R}]), (7.15)

[ω, q^{R}] is bounded and hence [H_{0}, Γ(q^{R})](H_{0} + 1)^{−1} is bounded. Therefore, Γ(q^{R}) preserves D(H_{0}).

Using that on D(H_{0}) ∩ D(N^{n}) we have H = H_{0} + V and Γ(q^{R}) preserves D(H_{0}) ∩ D(N^{n}), the following identity is valid as an operator identity on D(H_{0}) ∩ D(N^{n}):

[H, Γ(q^{R})] = [H_{0}, Γ(q^{R})] + [V, Γ(q^{R})] =: T.

Using (7.15) and the fact that [ω, q^{R}] ∈ O(R^{−1}), we get

[Γ(q^{R}), H_{0}] ∈ (N + 1)O_{N}(R^{−1}),

and using Lemma 3.17 and Lemma 6.3, we have

[Γ(q^{R}), V] ∈ (N + 1)^{n}O_{N}(R^{−s}),

which gives T ∈ (N + 1)^{n}O(R^{− inf(s,1)}). (7.16)

Now let

\begin{aligned}R(z) & := [Γ(q^{R}), (z − H)^{−1}] \\ & = −(z − H)^{−1}[Γ(q^{R}), H](z − H)^{−1}. \end{aligned}

We want to show that

N^{m}R(z)(H + b)^{−n−m} ∈ |Imz|^{−2}O(R^{− inf(s,1)}), m ≥ 0, (7.17)

(H + b)^{−n−m}R(z)N^{m} ∈ |Imz|^{−2}O(R^{− inf(s,1)}), m ≥ 0. (7.18)

By the higher order estimates (7.1) D(H^{n}) ⊂ D(H_{0}) ∩ D(N^{n}), so the following operator identity holds on D(H^{n−1}):

R(z) = (z − H)^{−1}T (z − H)^{−1}.

Now

\begin{matrix}∥N^{m}R(z)(H + b)^{−n−m}∥ ≤ ∥N^{m}(z − H)^{−1}(N + 1)^{−m+1}∥ \\ ×∥(N + 1)^{m−1}T (N + 1)^{−n−m+1}∥∥(N + 1)^{n+m−1}(H + b)^{−n−m+1}∥∥(H + b)(z − H)^{−1}∥, \end{matrix}

which proves (7.17), and (7.18) follows then by taking the adjoint.

If χ ∈ C_{0}^{∞}(IR), we denote by χ̃ ∈ C_{0}^{∞}(C) an almost analytic extension of χ, satisfying

\begin{matrix}χ̃_{|IR} = χ, \\ |∂_{\overline{z}}χ̃(z)| ≤ C_{n}|Imz|^{n}, n ∈ IN. \end{matrix}

We use the following functional calculus formula (see [HS, DG2]) for χ ∈ C_{0}^{∞}(IR):

χ(A) = \frac{i}{2π} ∫_{C} ∂_{\overline{z}}χ̃(z)(z − A)^{−1}dz ∧ d \overline{z}. (7.19)

Let now χ_{1} ∈ C_{0}^{∞}(IR) with χ_{1}χ = χ and χ̃_{1} an almost analytic extension of χ_{1}. We write:

N^{m}[χ(H), Γ(q^{R})]N^{p}
= N^{m}χ_{1}(H)[χ(H), Γ(q^{R})]N^{p} + N^{m}[χ_{1}(H), Γ(q^{R})]χ(H)N^{p}
= \frac{i}{2π} ∫_{C} ∂_{\overline{z}}χ̃(z)N^{m}χ_{1}(H)R(z)N^{p}dz ∧ d \overline{z}
+\frac{i}{2π} ∫_{C} ∂_{\overline{z}}χ̃_{1}(z)N^{m}R(z)χ(H)N^{p}dz ∧ d \overline{z}.

Using Prop. 7.10 ii), (7.17) and (7.18), we see that N^{m}[χ(H), Γ(q^{R})]N^{p} is O(R^{− inf(s,1)}), as claimed. □

The following lemma is analogous to [DG1, Lemma 3.4].

Lemma 7.12 Let j_{0} ∈ C_{0}^{∞}(IR^{d}), j_{∞} ∈ C^{∞}(IR^{d}), 0 ≤ j_{0}, 0 ≤ j_{∞}, j_{0}^{2} + j_{∞}^{2} ≤ 1, j_{0} = 1 near 0 (and hence j_{∞} = 0 near 0). Set j := (j_{0}, j_{∞}) and for R ≥ 1 j^{R} = (j_{0}^{R}, j_{∞}^{R}). Then for χ ∈ C_{0}^{∞}(IR):

χ(H^{ext})I^{∗}(j^{R}) − I^{∗}(j^{R})χ(H) ∈ \{ \begin{matrix}(N + 1)^{−∞}Ǒ(R^{− inf(s,1)}) under hypothesis (Is), \\ (N + 1)^{−∞}ǒ(R^{0}) under hypothesis (A). \end{matrix}

Proof. Again we will only prove the lemma under hypothesis (Is). We have by Lemma 3.11 i) H_{0}^{ext}I^{∗}(j^{R}) − I^{∗}(j^{R})H_{0} ∈ (N + 1)Ǒ_{N}(R^{−1}). (7.20) This implies that I^{∗}(j^{R}) sends D(H_{0}) into D(H_{0}^{ext}), and since I^{∗}(j^{R})N = (N_{0} + N_{∞})I^{∗}(j^{R}), I^{∗}(j^{R}) sends also D(N^{n}) into D((N_{0} + N_{∞})^{n}).

Next by Lemma 3.17 and Lemma 6.3 we obtain (V ⊗ 1l)I^{∗}(j^{R}) − I^{∗}(j^{R})V ∈ (N + 1)^{n}Ǒ_{N}(R^{−s}). (7.21) This and (7.20) show that as an operator identity on D(H_{0}) ∩ D(N^{n}) we have H^{ext}I^{∗}(j^{R}) − I^{∗}(J^{R})H ∈ (N + 1)^{n}Ǒ_{N}(R^{− min(1,s)}). (7.22) Using then the higher order estimates (7.1) and the fact that I^{∗}(j^{R}) sends D(H_{0}) into D(H_{0}^{ext}) and D(N^{n}) into D((N_{0} + N_{∞})^{n}), we obtain the following operator identity on D(H^{n}):

R(z) = (z − H^{ext})^{−1}I^{∗}(j^{R}) − I^{∗}(j^{R})(z − H)^{−1}
= (z − H^{ext})^{−1}(I^{∗}(j^{R})H − H^{ext}I^{∗}(j^{R}))(z − H)^{−1}.

Using Prop. 7.10, we see that (N_{0} + N_{∞})^{m}R(z)(H + b)^{−m−n} ∈ O(|Imz|^{−2})R^{− inf(s,1)}, (7.23) (H^{ext} + b)^{−m−n}R(z)N^{m} ∈ O(|Imz|^{−2})R^{− inf(s,1)}. (7.24) Let us again pick χ_{1} ∈ C_{0}^{∞}(IR) with χ_{1}χ = χ. We have:

(N_{0} + N_{∞})^{m}χ(H^{ext})I^{∗}(j^{R}) − I^{∗}(j^{R})χ(H)N^{m}

= (N_{0} + N_{∞})^{m}χ_{1}(H^{ext})(χ(H^{ext})I^{∗}(j^{R}) − I^{∗}(j^{R})χ(H))N^{m}
+(N_{0} + N_{∞})^{m}(χ_{1}(H^{ext})I^{∗}(j^{R}) − I^{∗}(j^{R})χ_{1}(H))χ(H)N^{m}
= \frac{i}{2π} ∫_{C} ∂_{\overline{z}}χ̃(z)(N_{0} + N_{∞})^{m}χ_{1}(H^{ext})R(z)N^{m}dz ∧ d \overline{z}
+\frac{i}{2π} ∫_{C} ∂_{\overline{z}}χ̃_{1}(z)(N_{0} + N_{∞})^{m}R(z)χ(H)N^{m}dz ∧ d \overline{z}.

Using Prop. 7.10 ii), (7.23) and (7.24), the above operator is O(R^{− inf(s,1)}), as claimed. □

8 A conjugate operator for P (φ)_{2} Hamiltonians

8.1 Introduction

This section is devoted to the study of a conjugate operator for P (φ)_{2} Hamiltonians. The central point of the construction of a conjugate operator A for a Hamiltonian H is the proof that the quadratic form [H, iA] defined on D(H) ∩ D(A) extends as a bounded operator from D(H) to D(H)^{∗} which is locally positive, ie such that 1l_{Δ}(H)[H, iA]1l_{Δ}(H) ≥ c_{0}1l_{Δ}(H) + K, for c_{0} > 0, Δ ⊂ IR an open interval and K a compact operator. However the local positivity of the quadratic form [H, iA] is not sufficient to apply the Mourre method. Additional conditions on H, A are needed. It seems that the weakest property one can impose is the property that H is class C^{1}(A), introduced in the book [ABG].

Let us recall the precise definition. First let us define this property for a bounded operator B on H. Let A be a self-adjoint operator on H. We say that B ∈ C^{1}(A) if the map

s ↦ e^{isA}Be^{−isA}

is C^{1} for the strong topology. The condition that B ∈ C^{1}(A) can be characterized in terms of the commutator [B, A]. Namely (see [ABG, Lemma 6.2.9]), B ∈ C^{1}(A) if and only if the quadratic form on D(A)

Q(v, v) = (Av, Bv) − (B^{∗}v, Av), v ∈ D(A)

satisfies |Q(v, v)| ≤ C∥v∥^{2}, v ∈ D(A). (8.1)

Let us note the following consequence of the C^{1}(A) property, proven in [ABG]:

Proposition 8.1 Let B be bounded and B ∈ C^{1}(A). Then B maps D(A) into itself, so that the expression [A, iB] makes sense as an operator on D(A) and

\frac{d}{ds}e^{isA}Be^{−isA} = [A, iB].

For an arbitrary self-adjoint operator H [ABG] propose a different definition (which is equivalent to the one above for bounded H): H ∈ C^{1}(A) if for some z ∉ σ(H) the map

s ↦ e^{isA}(z − H)^{−1}e^{−isA}

is C^{1} for the strong topology. These are some of the consequences of the fact that H ∈ C^{1}(A) (see [ABG, Thm. 6.2.10, Prop. 7.2.10]):

Proposition 8.2 Let A, H be self-adjoint and H ∈ C^{1}(A). Then the following is true:

i) For z ∉ σ(H), (z − H)^{−1} maps D(A) into itself, the space (z − H)^{−1}D(A) does not depend on z and is a dense subspace of D(A) ∩ D(H). ii) D(A) ∩ D(H) is dense both in D(A) and in D(H). iii) The quadratic form [H, iA] on D(A) ∩ D(H) extends uniquely to a bounded operator [H, iA]_{0} from D(H) to D(H)^{∗}. iv) For any λ ∈ IR, the virial relation 1l_{{λ}}(H)[H, iA]_{0}1l_{{λ}}(H) = 0 holds.

In our case the application of the Mourre method to P (φ)_{2} Hamiltonians runs into two related problems. The first problem is that (except for the ϕ_{2}^{4} model) the domain of H is not explicitly known. (This indicates that it is unlikely that the stronger set of conditions introduced in the original paper of Eric Mourre [Mo], which require in particular that e^{isA} preserves D(H), can be checked for P (φ)_{2} Hamiltonians).

The second problem is that the actual computation of [H, iA], needed to prove its positivity, cannot be done easily on D(A) ∩ D(H), since the identity H = H_{0} + V needed to do this computation does not make sense on D(H). For general P (φ)_{2} Hamiltonians, these two problems will be addressed in Thms. 8.4 and 8.7 below. If degP = 4, a simpler argument, using the Wick calculus instead of the Q−representation can be used to show that H ∈ C^{1}(A). This is done in Thm. 8.8

8.2 Analysis of [H, iA] part I

Let a = \frac{1}{2}(x · D_{x} + D_{x} · x) = −\frac{1}{2}(k · D_{k} + D_{k} · k) be the generator of dilations on L^{2}(IR, dk). We denote by A the second quantized generator of dilations

A := dΓ(a).

We put

H_{0}^{(1)} := [H_{0}, iA] as a quadratic form on D(A) ∩ D(H_{0}),

(8.2) V^{(1)} := [V, iA], as a quadratic form on D(A) ∩ D(N^{n}).

H^{(1)} := [H, iA], as a quadratic form on D(A) ∩ D(H^{n}).

Remark 8.3 It is important here to define the quadratic form [H, iA] on D(A) ∩ D(H^{n}). In fact from the higher order estimates (7.1), we know that D(H^{n}) ⊂ D(H_{0}) ∩ D(N^{n}) so that on D(A) ∩ D(H^{n}) we have [H, iA] = [H_{0}, iA] + [V, iA].

By a direct computation we see that H_{0}^{(1)} extends uniquely as a bounded operator from D(H_{0}) to H (still denoted by H_{0}^{(1)}), equal to dΓ(k · ∇ω(k)).

Note that the operator e^{isa} commutes with the conjugation c. Therefore, since V is a multiplication operator on Q-space, so is V^{(1)}. In this subsection we will study properties of V^{(1)} which follow from the assumption (M1) and its expression as a Wick polynomial.

It is convenient to introduce the notation H_{s} = e^{isA}He^{−isA}, H_{0,s} = e^{isA}H_{0}e^{−isA}, V_{s} = e^{isA}V e^{−isA}. We have H_{0,s} = dΓ(ω_{s}), for ω_{s}(k) = ω(e^{s}k), and using (3.18), we see that V_{s} is a Wick polynomial with the kernels w_{p,s} = Γ(e^{isa})w_{p}.

Theorem 8.4 Assume hypothesis (M1). Then

i) the form V^{(1)} extends to a bounded operator from D(N^{n}) to H. It is a multiplication operator on Q-space by a function in ∩_{p<∞} L^{p}(Q, dμ).

ii) the form H^{(1)} extends uniquely as an operator still denoted by H^{(1)} bounded from D(H^{n}) into H.

iii) for all z ∉ σ(H), (z − H)^{−1} is bounded from D(A) into itself.

iv) for all z ∉ σ(H), for r ≥ 2n, (z − H)^{−r} ∈ C^{1}(A) and hence the following identity holds as an identity between bounded operators from D(A) to H:

A(H − z)^{−r} = (H − z)^{−r}A + i\frac{d}{ds}(H_{s} − z)_{|s=0}^{−r}, (8.3)

where

\frac{d}{ds}(H_{s} − z)_{|s=0}^{−r} = ∑_{j=0}^{r−1}(H − z)^{−r+j}(H_{0}^{(1)} + V^{(1)})(H − z)^{−j−1} (8.4)

is a bounded operator on H.

In the next subsection we will need to approximate V by V_{κ}. Let us set

V_{κ}^{(1)} := [V_{κ}, iA], as a quadratic form on D(A) ∩ D(N^{n}).

Proposition 8.5 Assume hypothesis (M1). Then

i) the form V_{κ}^{(1)} extends to a bounded operator from D(N^{n}) to H. It is a multiplication operator on Q-space by a function in ∩_{p<∞} L^{p}(Q, dμ).

ii) as bounded operators from D(N^{n}) to H, we have:

V^{(1)} = lim_{κ→+∞} V_{κ}^{(1)}.

iii) For some ϵ > 0

∥V^{(1)} − V_{κ}^{(1)}∥_{L^{p}(Q,dμ)} ≤ C(p − 1)^{n}κ^{−ϵ}, ϵ > 0.

Lemma 8.6 Assume hypothesis (M1). Then

i) uniformly in κ,

dΓ(a)w_{p,κ} ∈ L^{2}(IR^{p}), 1 ≤ κ ≤ ∞,

ii) there exists ϵ > 0 such that:

∥dΓ(a)(w_{p,κ} − w_{p,∞})∥_{L^{2}(IR^{p})} ∈ O(κ^{−ϵ}).

Proof. We compute

\begin{matrix}dΓ(a)w_{p,∞} \\ = (aĝ(∑_{1}^{p} k_{i}))Π_{1}^{p}ω(k_{i})^{− \frac{1}{2}} + ∑_{1}^{p} ĝ(∑_{1}^{p} k_{i})(aω(k_{i})^{− \frac{1}{2}} ))Π_{j≠i}ω(k_{j})^{− \frac{1}{2}} , \end{matrix}

and

\begin{matrix}dΓ(a)w_{p,κ} \\ = (aĝ(∑_{1}^{p} k_{i}))Π_{1}^{p}χ̂(\frac{k_{i}}{κ})ω(k_{i})^{− \frac{1}{2}} + ∑_{1}^{p} ĝ(∑_{1}^{p} k_{i})(aω(k_{i})^{− \frac{1}{2}} χ̂(\frac{k_{i}}{κ}))Π_{j≠i}ω(k_{j})^{− \frac{1}{2}} χ̂(\frac{k_{j}}{κ}). \end{matrix}

Using (M1) and the bound (6.9), one sees that dΓ(a)w_{p,κ} ∈ L^{2}(IR^{p}), 1 ≤ κ ≤ ∞.

Then one checks that

(8.5) |a^{α}(1 − χ̂(\frac{k}{κ}))ω(k)^{− \frac{1}{2}} | ≤ C|k|^{− \frac{1}{2} +ϵ}κ^{−ϵ}, for some ϵ > 0 and α = 0, 1.

Using again (6.9) we deduce from (8.5) statement ii) in the lemma. □

Proof of Theorem 8.4. Applying Prop. 3.13, we obtain the following identity between quadratic forms on D(A) ∩ D(N^{n}):

\begin{aligned}V_{p}^{(1)} & := [∫ g(x) :φ(x)^{p}: dx, iA] \\ & = ∑_{r=0}^{p} ( \begin{matrix}p \\ r \end{matrix} ) ∫ w_{p}^{(1)}(k_{1}, …, k_{r}, k_{r+1}, …, k_{p}) \\ & ×a^{∗}(k_{1}) ⋯ a^{∗}(k_{r})a(−k_{r+1}) ⋯ a(−k_{p})dk_{1} ⋯ dk_{p}, \end{aligned} (8.6)

where

w_{p}^{(1)} = dΓ(a)w_{p}.

By Lemma 8.6 with κ = ∞, w_{p}^{(1)} ∈ L^{2}(IR^{p}). Hence, the rhs of (8.6) defines a bounded operator from D(N^{n}) to H. Next we note that V^{(1)}Ω ∈ ⊕_{p=0}^{2n} ⊗_{s}^{p} 𝔥, since V^{(1)} is a Wick polynomial of degree 2n. Hence by Lemma 5.12 V^{(1)}Ω ∈ ∩_{p<∞} L^{p}(Q, dμ). Therefore, as a multiplication

operator V^{(1)} ∈ ∩_{p<∞} L^{p}(Q, dμ). This ends the proof of i).

Let us prove iv). For r ∈ IN, z ∈ C\σ(H), the following identity makes sense (all terms are bounded operators):

(H_{s} − z)^{−r} − (H − z)^{−r} = ∑_{j=0}^{r−1}(H_{s} − z)^{−r+j}(H − H_{s})(H − z)^{−j−1}.

We deduce easily from the explicit form of H_{0,s} that ∥H_{0,s}(H_{0}+1)^{−1}∥ ≤ C, uniformly for |s| ≤ 1. This implies that D(H_{0,s}^{r}) = D(H_{0}^{r}), r ∈ IN. Since on the other hand e^{isA} preserves N, we have

(N + 1)^{α}(H_{0} + 1)^{2} ≤ C(H_{s} + b)^{2n+α}, α ≥ 0, |s| ≤ 1, (8.7)

ie Rosen’s higher order estimates are uniformly valid for H_{s}, |s| ≤ 1. Hence if j ≥ n, (H_{s} − z)^{−j} sends H into D(H_{0}) ∩ D(N^{n}), which implies that for r ≥ 2n, 0 ≤ j ≤ r − 1:

(H_{s} − z)^{−r+j}(H_{s} − H)(H − z)^{−j−1} = (H_{s} − z)^{−r+j}(H_{0,s} − H_{0} + V_{s} − V)(H − z)^{−j−1}.

We note that (is)^{−1}(H_{0,s} − H_{0}) = dΓ((is^{−1})(ω_{s} − ω)) and that (is)^{−1}(H_{0,s} − H_{0})(H_{0} − z)^{−1} converges strongly to H_{0}^{(1)}(H_{0} − z)^{−1} when s → 0 and is uniformly bounded for |s| ≤ 1. Using (8.7) this shows that

(8.8) (is)^{−1}(H_{s} − z)^{−r+j}(H_{0,s} − H_{0})(H − z)^{−j−1} →^{strongly} (H − z)^{−r+j}H_{0}^{(1)}(H − z)^{−j−1}

and is uniformly bounded for |s| ≤ 1.

Next it follows from Lemma 8.6 i) for κ = ∞ that the map

s ↦ w_{p,s} = ĝ (e^{s} ∑_{i=1}^{p} k_{i}) Π_{i=1}^{p}ω(e^{s}k_{i})^{− \frac{1}{2}}

is C^{1}(IR, L^{2}(IR^{p})) with derivative dΓ(a)w_{p}. This implies that

(is)^{−1}(N + 1)^{−r_{1}}(V_{s} − V)(N + 1)^{−r_{2}} → i(N + 1)^{−r_{1}}V^{(1)}(N + 1)^{−r_{2}} (8.9)

in operator norm when s → 0, for r_{1} + r_{2} ≥ n. Using then the estimates (8.7), we deduce from (8.9) that

(is)^{−1}(H_{s} − z)^{−r+j}(V_{s} − V)(H − z)^{−j−1}

is uniformly bounded for |s| ≤ 1 and converges strongly to −i(H − z)^{−r+j}V^{(1)}(H − z)^{−j−1} when s → 0. This ends the proof of (8.4) and of iv).

By Prop. 8.2 we conclude that r ≥ 2n, (H − z)^{−r} is bounded on D(A). Since this is true for r = 0, this holds for any r ≥ 0 by interpolation, which shows iii).

Let us prove ii). By Remark 8.3 the quadratic form [H, iA] on D(A) ∩ D(H^{n}) is equal to [H_{0}, iA]+[V, iA]. We have seen that [H_{0}, iA] extends as an operator H_{0}^{(1)} such that H_{0}^{(1)}(H_{0}+1)^{−1} is bounded, which by the higher order estimates implies that H_{0}^{(1)}(H + b)^{−n} is bounded. In i) we have seen that [V, iA] extends as an operator V^{(1)} such that V^{(1)}(N +1)^{−n} is bounded, which again by the higher order estimates implies that V^{(1)}(H + b)^{−n} is bounded. It remains to check that the extension of [H, iA] to an operator H^{(1)} with domain D(H^{n}) is unique, ie that D(A) ∩ D(H^{n}) is dense in D(H^{n}). For u = (H + b)^{−n}v ∈ D(H^{n}), we set u_{ϵ} = (H + b)^{−n}(1 + iϵA)^{−1}v. Clearly u_{ϵ} ∈ D(H^{n}) and u_{ϵ} tends to u in D(H^{n}) when ϵ → 0. It follows then from iii) that u_{ϵ} ∈ D(A), which completes the proof of ii). □

Proof of Proposition 8.5. i) follows from Lemma 8.6.

To prove ii) we note that

∥(V_{p}^{(1)} − V_{p,κ}^{(1)})(N + 1)^{−n}∥ ≤ C∥w_{p} − w_{p,κ}∥_{L^{2}(IR^{p})} ≤ Cκ^{−ϵ}.

To show iii) we note that

∥V_{p}^{(1)} − V_{p,κ}^{(1)}∥_{L^{2}(Q,dμ)} = ∥w_{p} − w_{p,κ}∥_{L^{2}(IR^{p})} ≤ Cκ^{−ϵ}.

Then we use Lemma 5.12. □

8.3 Analysis of [V, iA] part II

In this subsection we continue our study of [H, iA]. The main new ingredient is the use of hypothesis (B1), which will allow to dominate |V^{(1)}|, as a function on Q−space, by H, using hypercontractivity arguments.

Theorem 8.7 Assume hypothesis (B1).

i) There exists c_{0}, b > 0 such that

|(u, H^{(1)}u)| ≤ c_{0}(u, (H + b)u), u ∈ D(H^{n}),

and hence H^{(1)} extends uniquely as an operator bounded from D(H^{\frac{1}{2}} ) to D(H^{\frac{1}{2}} )^{∗}. ii) H ∈ C^{1}(A). iii) The operator [H, iA]_{0} from D(H) into D(H)^{∗}, associated to [H, iA] by Proposition 8.2, coincides with H^{(1)} and hence is bounded from D(H^{\frac{1}{2}} ) to D(H^{\frac{1}{2}} )^{∗}. iv) The virial relation holds

1l_{{λ}}(H)[H, iA]_{0}1l_{{λ}}(H) = 0, λ ∈ IR.

Thm. 8.7 is the main result of this section. Property ii) allows in particular to justify the virial relation iv). Property iii) allows to actually compute [H, iA]_{0} which will be important to prove its positivity in Subsect. 9.2.

In the φ_{2}^{4} case a similar result holds under weaker hypotheses on g.

Theorem 8.8 Assume degP = 4 and hypotheses (M1), (C).

i) There exists c_{0}, b > 0 such that

|(Hu, Au) − (Au, Hu)| ≤ c_{0}∥(H + b)u∥^{2}, u ∈ D(A) ∩ D(H),

and hence H^{(1)} extends uniquely as an operator bounded from D(H) to D(H)^{∗}. ii) H ∈ C^{1}(A). iii) The operator [H, iA]_{0} from D(H) into D(H)^{∗} associated to [H, iA] coincides with H^{(1)}. iv) The virial relation holds

1l_{{λ}}(H)[H, iA]_{0}1l_{{λ}}(H) = 0, λ ∈ IR.

It is convenient to make a specific choice of the cutoff χ used to define the cutoff interactions V_{κ}. Namely we will fix for this section

χ(x) = \frac{1}{2}e^{−|x|}, χ̂(k) = (1 + k^{2})^{−1}. (8.10)

We denote by u ⋆ v the convolution

u ⋆ v(x) = ∫ u(y)v(x − y)dy,

so that F(u ⋆ v) = F(u)F(v). Recall that the function f_{κ} was defined in (6.4).

Lemma 8.9

τ_{x}(iaf_{κ}) = 2τ_{x}f_{κ} − α_{κ} ⋆ τ_{x}f_{κ},

where

α_{κ}(x) = κe^{−κ|x|} + \frac{m}{4}e^{−m|x|}.

Proof. After conjugation by the Fourier transformation, we are reduced to check that

(−k∂_{k} − \frac{1}{2})f̂_{κ} = (2δ_{0} − α̂_{κ})f̂_{κ}.

This is a direct computation, using (8.10). □

Lemma 8.10 Assume hypothesis (B0). Then there exists C_{0} such that for κ ≥ m:

|α_{κ} ⋆ g(x)| ≤ C_{0}g(x). (8.11)

An analogous estimate is true if we replace α_{κ}(x) with x∂_{x}α_{κ}(x) or α_{κ} ⋆ α_{κ}(x).

Proof. It is sufficient to show the estimate replacing α_{κ} with a function κψ(κx) for ψ ∈ S(IR). Now by (B0):

∫ g(x − x′)ψ(κx′)κdx′ ≤ Cg(x) ∫ ψ(\frac{x′}{κ})⟨x′⟩κdx′.

□ Lemma 8.11 We have:

\begin{matrix}[A, i ∫ g(x)φ_{κ}(x)^{p}dx] \\ = ∫ (2p + ∂_{x}x)g(x)φ_{κ}(x)^{p}dx \\ −p ∫ ∫ g(x)α_{κ}(x′ − x)φ_{κ}(x′)(φ_{κ}(x))^{p−1}dxdx′. \end{matrix} (8.12)

Proof. Using φ_{κ}(x) = ϕ(τ_{x}f_{κ}) (8.13)

we obtain as a quadratic form on D(A) ∩ D(N^{\frac{1}{2}} ):

[A, iφ_{κ}(x)] = ϕ(iaτ_{x}f_{κ}).

Now

iaτ_{x}f_{κ}(y) = y∂_{y}f_{κ}(y − x) + \frac{1}{2}f_{κ}(y − x)
= (y − x)∂_{y}f_{κ}(y − x) + \frac{1}{2}f_{κ}(y − x) − x∂_{x}f_{κ}(y − x)
= (2 − x∂_{x})τ_{x}f_{κ}(y) − ∫ α_{κ}(x′ − x)τ_{x′}f_{κ}(y)dx′,

by Lemma 8.9. This gives

[A, iφ_{κ}(x)] = (2 − x∂_{x})φ_{κ}(x) − ∫ α_{κ}(x′ − x)φ_{κ}(x′)dx′. (8.14)

Since ia = x∂_{x} + \frac{1}{2} preserves L_{IR}^{2}(IR, dx), we have [ϕ(iaτ_{x}f), ϕ(τ_{x}f)] = 0 and hence

\begin{aligned}[A, iφ_{κ}(x)^{p}] & = pφ_{κ}(x)^{p−1}[A, iφ_{κ}(x)] \\ & = 2pφ_{κ}(x)^{p} − x∂_{x}φ_{κ}^{p}(x) − p ∫ α_{κ}(x′ − x)φ_{κ}(x′)φ_{κ}^{p−1}(x)dx′, \end{aligned} (8.15)

as a quadratic form on D(A) ∩ D(N^{p/2}). The lemma follows then using (8.15) and integration by parts. □

As a consequence of Lemmas 8.9, 8.11, we have the following inequality, which should be understood as an inequality between functions on Q-space:

Lemma 8.12 Assume hypothesis (B1). Then for p ∈ IN

(8.16) |[A, i ∫ g(x)φ_{κ}(x)^{p}dx]| ≤ C_{p} ∫ g(x)|φ_{κ}(x)|^{p}dx, uniformly for κ ≥ m.

Proof. Let us denote by I_{1}, I_{2} the terms in the r.h.s. of (8.12). We will estimate separately I_{1} and I_{2}.

Estimate of I_{1}:

Since by (B1) |x∂_{x}g(x)| ≤ Cg(x), we see that

|I_{1}| ≤ C ∫ g(x)|φ_{κ}(x)|^{p}dx. (8.17)

Estimate of I_{2}:

We have:

|I_{2}| = p| ∫ g(x)α_{κ}(x′ − x)φ_{κ}(x′)φ_{κ}(x)^{p−1}dxdx′|
≤ C ∫ g(x)|α_{κ}(x′ − x)||φ_{κ}(x′)|^{p}dxdx′
+C ∫ g(x)|α_{κ}(x′ − x)||φ_{κ}(x)|^{p}dxdx′,

using the fact that ab^{p−1} ≤ C_{p}(a^{p} + b^{p}). This yields

|I_{2}| ≤ C_{0}c_{κ} ∫ g(x)|φ_{κ}(x)|^{p}dx + C_{0} ∫ g_{κ}(x)|φ_{κ}(x)|^{p}dx,

for

c_{κ} = ∫ α_{κ}(x′)dx′, g_{κ} = g ⋆ α_{κ}.

We have used here the fact that g and α_{κ} are positive functions. Clearly

(8.18) |c_{κ}| ≤ C, uniformly in κ,

and from Lemma 8.10, we get that

(8.19) |g_{κ}(x)| ≤ C_{0}g(x), uniformly in κ ≥ m.

From (8.18) and (8.19), we obtain:

(8.20) |I_{2}| ≤ C_{0} ∫ g(x)|φ_{κ}(x)|^{p}dx, uniformly in κ ≥ m.

□

Proposition 8.13 Assume hypotheses (B1). Then there exists c > 0 such that for t > 0:

e^{−t(cV −|V^{(1)}|)} ∈ L^{1}(Q, dμ).

Proof. Set

W := cV − |V^{(1)}|.

To check that e^{−tW} ∈ L^{1}(Q, dμ), we use Lemma 5.9. The first bound of (5.4) follows from Prop. 8.5 iii). Let us now check the second bound.

Using Wick identities (5.2) and Lemma 8.12 we obtain

|V_{κ}^{(1)}| ≤ C_{0} ∑_{p=0}^{2n} |a_{p}| ∑_{r=0}^{[p/2]} \frac{p!2^{−r}}{(p − 2r)!r!}∥φ_{κ}(x)Ω∥^{2r} ∫ g(x)|φ_{κ}(x)|^{p−2r}dx,

so that

cV_{κ} − V_{κ}^{(1)} ≥ ∫ g(x)F_{κ}(φ_{κ}(x))dx,

where F_{κ}(λ) is a function as in (6.15), with c_{2n,0} = ca_{2n} − c_{0}|a_{2n}| > 0 for c > c_{0}. Using (6.16) we obtain cV_{κ} − |V_{κ}^{(1)}| ≥ −c_{2}∥g∥_{L^{1}(IR)}(∥φ_{κ}(x)Ω∥^{2n} + 1), (8.21) which since ∥φ_{κ}(x)Ω∥ = O(ln κ^{\frac{1}{2}} ) completes the proof of the second bound in (5.4). Applying now Lemma 5.9, we get that there exists c large enough such that e^{−tW} ∈ L^{1}(Q, dμ) for all t > 0. This completes the proof of the proposition. □

Proof of Thm. 8.7. For large enough C we have Cω −ω^{(1)} ≥ C_{0} > 0. Therefore, we can apply Thm. 5.11 to a = Cω − ω^{(1)} and W = CV − |V^{(1)}| and show that CH_{0} − H_{0}^{(1)} + CV − |V^{(1)}| is bounded from below on D(CH_{0} − H_{0}^{(1)}) ∩ D(N^{n}). But D(CH_{0} − H_{0}^{(1)}) = D(H_{0}) and D(H_{0}) ∩ D(N^{n}) contains D(H^{n}) by the higher order estimates and hence is dense in D(H). Therefore the inequality

H_{0}^{(1)} + V^{(1)} ≤ C(H_{0} + V + b) on D(H^{n})

extends as the inequality

H^{(1)} ≤ C(H + b) on D(H).

Likewise, we prove

−H^{(1)} ≤ C(H + b) on D(H).

This proves i).

Next, let us show ii). To prove that H ∈ C^{1}(A), we check condition (8.1). Since by Thm. 8.4 (H + b)^{−1} is bounded on D(A), we can write for v ∈ D(A):

\begin{matrix}Q(v, v) = (v, A(H + b)^{−1}v) − (A(H + b)^{−1}v, v) \\ = (Hu, Au) − (Au, Hu), for u = (H + b)^{−1}v. \end{matrix}

Since (H + b)^{−1}D(A) ⊂ D(A) ∩ D(H), (8.1) is implied by

|(Hu, Au) − (Au, Hu)| ≤ C(∥Hu∥^{2} + ∥u∥^{2}), u ∈ D(A) ∩ D(H). (8.22)

We know by i) that (8.22) holds for u ∈ D(A)∩D(H^{n}). So to prove (8.22) it suffices to show that D(A)∩D(H^{n}) is dense in D(A)∩D(H) for the intersection topology. Let hence u ∈ D(A)∩D(H) and u_{ϵ} = (1 + iϵH)^{−2n}u. Clearly u_{ϵ} ∈ D(A) ∩ D(H^{n}), u_{ϵ} → u, Hu_{ϵ} → Hu. Now from (8.3), we get:

\begin{matrix}A(1 + iϵH)^{−2n}u = (1 + iϵH)^{−2n}Au − iϵ ∑_{j=0}^{2n−1}(1 + iϵH)^{−2n+j}H^{(1)}(1 + iϵH)^{−j−1}u \\ =: (1 + iϵH)^{−2n}Au − R_{ϵ}u. \end{matrix}

We claim that

s- lim_{ϵ→0} R_{ϵ} = 0, (8.23)

which will imply that Au_{ϵ} → Au when ϵ → 0. In fact we write

\begin{matrix}(1 + iϵH)^{−2n+j}H^{(1)}(1 + iϵH)^{−j−1} \\ = (1 + iϵH)^{−2n+j}(H + b)^{\frac{1}{2}} (H + b)^{− \frac{1}{2}} H^{(1)}(H + b)^{− \frac{1}{2}} (H + b)^{\frac{1}{2}} (1 + iϵH)^{−j−1}. \end{matrix}

Using i) and the bound ∥(1 + iϵH)^{−j}(H + b)^{\frac{1}{2}} ∥ ∈ O(ϵ^{− \frac{1}{2}} ) for j ≥ 1, we get that R_{ϵ} ∈ O(1). So it suffices to prove (8.23) on a dense subset of H. For u ∈ D(H^{n}), we have:

(1 + iϵH)^{−r+j}H^{(1)}(1 + iϵH)^{−j−1}u = (1 + iϵH)^{−r+j}H^{(1)}(H + b)^{−n}(1 + iϵH)^{−j−1}(H + b)^{n}u, which by Thm. 8.4 ii) shows that R_{ϵ}u → 0 when ϵ → 0. This proves that H is of class C^{1}(A).

To prove iii), we note that both [H, iA]_{0} and H^{(1)} are extensions of [H, iA] on D(A) ∩ D(H) and D(A) ∩ D(H^{n}) respectively. Since D(A) ∩ D(H^{n}) is dense in D(A) ∩ D(H) these two extensions coincide.

Finally the fact that the virial theorem is true follows also from the C^{1}(A) property (see [ABG, Prop. 7.2.10]). □

Proof of Thm. 8.8. We will prove that

(Hu, Au) − (Au, Hu) = i^{−1}(u, H_{0}^{(1)}u) + (u, V^{(1)}u), u ∈ D(H) ∩ D(A). (8.24)

We have seen in Subsect. 8.2 that

|(u, H_{0}^{(1)}u)| ≤ C∥H0^{\frac{1}{2}} u∥^{2}, u ∈ D(H_{0}),

and

|(u, V^{(1)}u)| ≤ ∥(N + 1)u∥^{2}, u ∈ D(N), since degP = 4.

Hence i) follows from (8.24) and the higher order estimates. Hence we have shown (8.22) which, as we have seen in the proof of Thm. 8.7, implies that H ∈ C^{1}(A), ie that ii) holds. Finally iv) follows from ii) and iii) follows from (8.24). So it suffices to prove (8.24).

Since by Thm. 7.2 D(H) = D(H_{0}) ∩ D(V), we have:

(Hu, Au) − (Au, Hu) = (H_{0}u, Au) − (Au, H_{0}u) + (V u, Au) − (Au, V u), u ∈ D(H) ∩ D(A).

By definition

(H_{0}u, Au) − (Au, H_{0}u) = i^{−1}(u, H_{0}^{(1)}u),

so it remains to justify the identity

(V u, Au) − (Au, V u) = i^{−1}(u, V^{(1)}u), u ∈ D(H) ∩ D(A). (8.25)

Note that while (8.25) holds for example on D(N^{2}) ∩ D(A), it is not obvious that it extends to u ∈ D(H) ∩ D(A). In fact the expression of V as a Wick polynomial of order 4 needed to prove (8.25) is meaningful on D(N^{2}), but not on D(V). To justify (8.25) we use an approximation argument similar to the one used in the proof of Thm. 8.3.

So let u ∈ D(H) ∩ D(A), and u_{ϵ} = (1 + iϵN)^{−1}u. Since by the higher order estimates D(H) ⊂ D(N), u_{ϵ} ∈ D(N^{2}) ∩ D(A) and hence (8.25) holds for u_{ϵ}. So it remains to show that Au_{ϵ} → Au, V u_{ϵ} → V u when ϵ → 0. The first convergence is obvious. To prove the second one it suffices to show that

V (1 + iϵN)^{−1}u − (1 + iϵN)^{−1}V u → 0, when ϵ → 0. (8.26)

Note that since u ∈ D(N), we can write V as a Wick polynomial to prove (8.26). If

W = ∫ w(k_{1}, ⋯, k_{4})a^{∗}(k_{1}) ⋯ a^{∗}(k_{r})a(−k_{r+1}) ⋯ a(−k_{4})dk_{1} ⋯ dk_{4},

for w ∈ L^{2}(IR^{4}), then

\begin{matrix}W (1 + iϵN)^{−1} \\ = ∫ (1 + iϵN + iϵ(4 − 2r))^{−1}w(k_{1}, ⋯, k_{4})a^{∗}(k_{1}) ⋯ a^{∗}(k_{r})a(−k_{r+1}) ⋯ a(−k_{4})dk_{1} ⋯ dk_{4}. \end{matrix}

Using the first resolvent formula and the bound ∥N(1 + iϵN)^{−1}∥ ≤ ϵ^{−1}, we obtain that

\begin{matrix}∥(W (1 + iϵN)^{−1} − (1 + iϵN)^{−1}W)u∥ ≤ C∥(N + 1)u∥, \\ ∥(W (1 + iϵN)^{−1} − (1 + iϵN)^{−1}W)u∥ ≤ Cϵ∥(N + 1)^{2}u∥. \end{matrix} (8.27)

The first inequality in (8.27), it suffices to prove (8.26) for u in a dense subspace of D(N). By the second inequality in (8.27), (8.26) holds for u ∈ D(N^{2}).□

8.4 Analysis of [[H, iA], iA].

The aim of this subsection is to show that, under hypothesis (B2), H ∈ C^{2}(A). The structure of the argument is parallel to the arguments used in Subsects 8.2 and 8.3.

We put

H_{0}^{(2)} := [H_{0}^{(1)}, iA] as a quadratic form on D(A) ∩ D(H_{0}),

(8.28) V^{(2)} := [V^{(1)}, iA], as a quadratic form on D(A) ∩ D(N^{n}),

H^{(2)} := [H^{(1)}, iA], as a quadratic form on D(A) ∩ D(H^{n}).

By a direct computation we see that H_{0}^{(2)} extends uniquely as a bounded operator from D(H_{0}) to H (still denoted by H_{0}^{(2)}), equal to dΓ((k · ∇)^{2}ω(k)).

Proposition 8.14 Assume hypothesis (M2). Then

i) the form V^{(2)} extends to a bounded operator from D(N^{n}) to H. It is a multiplication operator on Q-space by a function in ∩_{p<∞} L^{p}(Q, dμ).

ii) the form H^{(2)} extends uniquely to a bounded operator from D(H^{n}) to H.

Proof. The proof is analogous to the proof of Thm. 8.4 i) and ii). We write V^{(2)} as a Wick polynomial using the fact that a^{2}g ∈ L^{2}(IR), which follows from hypothesis (M2). □

Then we set

V_{κ}^{(2)} := [V_{κ}^{(1)}, iA] as a quadratic form on D(A) ∩ D(N^{n}).

The following proposition is analogous to Prop. 8.5.

Proposition 8.15 Assume hypothesis (M2). Then

i) the form V_{κ}^{(2)} extends to a bounded operator from D(N^{n}) to H. It is a multiplication operator on Q-space by a function in ∩_{p<∞} L^{p}(Q, dμ).

ii) As bounded operators from D(N^{n}) to H, we have:

V^{(2)} = lim_{κ→+∞} V_{κ}^{(2)}.

iii) For some ϵ > 0

∥V^{(2)} − V_{κ}^{(2)}∥_{L^{p}(Q,dμ)} ≤ C(p − 1)^{n}κ^{−ϵ}, ϵ > 0.

Theorem 8.16 Assume hypothesis (B2).

i) There exists c_{0}, b > 0 such that

|(u, H^{(2)}u)| ≤ c_{0}(u, (H + b)u), u ∈ D(H^{n}).

Hence H^{(2)} extends uniquely to a bounded operator from D((H + b)^{\frac{1}{2}} ) to D((H + b)^{\frac{1}{2}} )^{∗}. ii) H ∈ C^{2}(A).

The proof of this theorem will be similar to the proof of Theorem 8.7. The main difference is the following lemma, which is used instead of Lemma 8.12.

Lemma 8.17 Assume hypothesis (B2). Then for p ∈ IN:

|[A, i[A, i ∫ g(x)φ_{κ}(x)^{p}dx]]| ≤ C ∫ g(x)|φ_{κ}(x)|^{p}dx, uniformly for κ ≥ m.

Proof. We recall from Lemma 8.11 that [A, i ∫ g(x)φ_{κ}(x)^{p}dx] = I_{1} + I_{2}, for

\begin{matrix}I_{1} = ∫ (2pg(x) + ∂_{x}xg(x))φ_{κ}(x)^{p}dx, \\ I_{2} = −p ∫ ∫ g(x)α_{κ}(x′ − x)φ_{κ}(x′)(φ_{κ}(x))^{p−1}dxdx′, \end{matrix}

The terms [A, iI_{1}] is completely analogous to [A, i ∫ g(x)φ_{κ}(x)^{p}dx], with g replaced by g_{1} = 2pg + ∂_{x}xg. It follows from hypothesis (B2) that |x∂_{x}g_{1}(x)| ≤ g(x). The argument used in the proof of Lemma 8.12 shows then that |[A, iI_{1}]| ≤ C ∫ g(x)|φ_{κ}(x)|^{p}dx.

We consider next [A, iI_{2}]. Using the identity (8.14), we obtain:

[A, iI_{2}] = p ∫ g(x)α_{κ}(x′ − x)x′∂_{x′}φ_{κ}(x′)φ_{κ}(x)^{p−1}dxdx′
+p ∫ g(x)α_{κ}(x′ − x)φ_{κ}(x′)x∂_{x}φ_{κ}(x)^{p−1}dxdx′
−4p ∫ g(x)α_{κ}(x′ − x)φ_{κ}(x′)φ_{κ}(x)^{p−1}dxdx′
−p ∫ g(x)α_{κ}(x′ − x)α_{κ}(x′′ − x′)φ_{κ}(x′′)φ_{κ}(x)^{p−1}dxdx′dx′′
−p(p − 1) ∫ g(x)α_{κ}(x′ − x)α_{κ}(x′′ − x)φ_{κ}(x′)φ_{κ}(x′′)φ_{κ}(x)^{p−2}dxdx′dx′′
= R_{1} + ⋯ + R_{5}.

(8.29)

The term R_{3} is equal to 4I_{2} and hence bounded by C ∫ g(x)|φ_{κ}(x)|^{p}dx. Integrating by parts, we have:

\begin{aligned}R_{1} + R_{2} & = −2p ∫ g(x)α_{κ}(x′ − x)φ_{κ}(x′)φ_{κ}(x)^{p−1}dxdx′ \\ & −p ∫ x∂_{x}g(x)α_{κ}(x′ − x)φ_{κ}(x′)φ_{κ}(x)^{p−1}dxdx′ \\ & −p ∫ g(x)(x′ − x)∂_{x′}α_{κ}(x′ − x)φ_{κ}(x′)φ_{κ}(x)^{p−1}dxdx′. \end{aligned} (8.30)

The first term in (8.30) equals 2I_{2}. The second term is similar to I_{2}, with g replaced by x∂_{x}g. Note that it follows from (B1) that |x∂_{x}g(x)| ≤ cg(x). The third term is also similar to I_{2}, with α_{κ} replaced by x∂_{x}α_{κ}. By the argument in Lemma 8.12, we obtain that R_{1} + R_{2} is bounded by C ∫ g(x)|φ_{κ}(x)|^{p}dx.

The term R_{4} is equal to

−p ∫ g(x)ρ_{κ}(x′′ − x)φ_{κ}(x′′)φ_{κ}(x)^{p−1}dxdx′′, for ρ_{κ} = α_{κ} ⋆ α_{κ}.

Again the argument in Lemma 8.12 shows that R_{4} is bounded by C ∫ g(x)|φ_{κ}(x)|^{p}dx. Finally to estimate R_{5}, we use the fact that abc^{p−2} ≤ C(a^{p} + b^{p} + c^{p}), and get:

R_{5} ≤ C ∫ g(x)α_{κ}(x′ − x)α_{κ}(x′′ − x)|φ_{κ}(x′′)|^{p}dxdx′dx′′
+C ∫ g(x)α_{κ}(x′ − x)α_{κ}(x′′ − x)|φ_{κ}(x′)|^{p}dxdx′dx′′
+C ∫ g(x)α_{κ}(x′ − x)α_{κ}(x′′ − x)|φ_{κ}(x)|^{p}dxdx′dx′′
≤ 2Cc_{κ} ∫ g_{κ}(x)|φ_{κ}(x)|^{p}dx + Cc_{κ}^{2} ∫ g(x)|φ_{κ}(x)|^{p}dx,

for c_{k} = ∫ α_{κ}(x)dx, g_{κ} = g ⋆ α_{κ}. Using (8.18), (8.19), we obtain that R_{5} is bounded by C ∫ g(x)|φ_{κ}(x)|^{p}dx. This completes the proof of the lemma. □

Proof of Theorem 8.16. i) is shown exactly as the analogous statement of Theorem 8.7, using Lemma 8.17 instead of Lemma 8.12.

Let us prove that H ∈ C^{2}(A). It follows first from the fact that H ∈ C^{1}(A) that the following identity holds as quadratic forms on D(A):

[(H + b)^{−1}, iA] = −(H + b)^{−1}H^{(1)}(H + b)^{−1},

(see [ABG, Thm. 6.2.10]). To show that H ∈ C^{2}(A), we have to check that

(u|[(H + b)^{−1}H^{(1)}(H + b)^{−1}, A]u) ≤ C∥u∥^{2}, u ∈ D(A). (8.31)

We have remarked in the proof of Theorem 8.4 (see (8.23)) that D(H^{n}) ∩ D(A) is dense in D(A) for the graph topology. So it is enough to show (8.31) for u ∈ D(H^{n}) ∩ D(A).

For u ∈ D(H^{n}) ∩ D(A), we have

\begin{matrix}((H + b)^{−1}H^{(1)}(H + b)^{−1}u, Au) − (Au, (H + b)^{−1}H^{(1)}(H + b)^{−1}u) \\ = (H^{(1)}(H + b)^{−1}u, (H + b)^{−1}Au) − ((H + b)^{−1}Au, H^{(1)}(H + b)^{−1}u) \\ = (H^{(1)}(H + b)^{−1}u, A(H + b)^{−1}u) − (A(H + b)^{−1}u, H^{(1)}(H + b)^{−1}u) \\ +i(H^{(1)}(H + b)^{−1}u, (H + b)^{−1}H^{(1)}(H + b)^{−1}u) \\ +i((H + b)^{−1}H^{(1)}(H + b)^{−1}u, H^{(1)}(H + b)^{−1}u). \end{matrix} (8.32)

We use the fact that H^{(1)}(H + b)^{−n} is bounded by Thm. 8.4 ii) to justify the first equality in (8.32). Then we note that since (H + b)^{−1} preserves D(A), the following identity is valid as bounded operators from D(A) to H:

(H + b)^{−1}A = A(H + b)^{−1} + i(H + b)^{−1}H^{(1)}(H + b)^{−1}. (8.33)

and we use the identity (8.33) in the second equality of (8.32).

Applying Thm. 8.7 i), we see that the last two terms of (8.32) are less than C∥u∥^{2}. This shows that as quadratic forms on D(H^{n}) ∩ D(A), we have:

[(H + b)^{−1}H^{(1)}(H + b)^{−1}, iA] = (H + b)^{−1}[H^{(1)}, iA](H + b)^{−1} + R,

where R is bounded for the topology of H. By i), also the first term on the rhs is bounded for the topology of H. □

9 Spectral analysis of P (φ)_{2} Hamiltonians

This section is devoted to the spectral theory of P (φ)_{2} Hamiltonians. We first show an HVZ type result. Note that the ⊂ part of the HVZ theorem is well known (see [GJ3], [S-H.K]), although our proof is different. We then prove the Mourre estimate, which implies the local finiteness of point spectrum and under additional hypotheses, the limiting absorption principle.

9.1 HVZ theorem and existence of a ground state

Theorem 9.1 We have

σ_{ess}(H) = [inf σ(H) + m, +∞[.

Consequently inf σ(H) is a discrete eigenvalue of H.

Let us pick functions j_{0}, j_{∞} ∈ C^{∞}(IR) with 0 ≤ j_{0} ≤ 1, j_{0} ∈ C_{0}^{∞}(IR), j_{0} = 1 near 0 and j_{0}^{2} + j_{∞}^{2} = 1. For R ≥ 1, j^{R} is defined as in Subsect. 7.5. We will also set q^{R} = (j_{0}^{R})^{2}.

Proof. We prove first the ⊂ part of the theorem. Let χ ∈ C_{0}^{∞}(] − ∞, inf σ(H) + m[). Because of suppχ we have:

χ(H^{ext}) = χ(H^{ext})1l_{{0}}(N_{∞}).

Hence using twice Lemma 7.12 we have

χ(H) = χ(H)I(j^{R})I^{∗}(j^{R}) = I(j^{R})χ(H^{ext})I^{∗}(j^{R}) + o(R^{0})
= I(j^{R})χ(H^{ext})1l_{{0}}(N_{∞})I^{∗}(j^{R}) + o(R^{0}) = I(j^{R})1l_{{0}}(N_{∞})I^{∗}(j^{R})χ(H) + o(R^{0}).

We claim that the operator I(j^{R})1l_{{0}}(N_{∞})I^{∗}(j^{R})χ(H) = Γ(q^{R})χ(H) is compact. In fact since (H_{0} + 1)^{\frac{1}{2}} (H + b)^{− \frac{1}{2}} is bounded by the first order estimates (6.11), it suffices to verify that Γ(q^{R})(H_{0} + 1)^{− \frac{1}{2}} is compact, which is easy (see eg [DG1, Lemma 4.2]). Hence Γ(q^{R})χ(H) is compact as a norm limit of compact operators.

Let us now prove the ⊃ part of the theorem. Note that it follows from the ⊂ part of the theorem that H admits a ground state. Let λ > inf σ(H) + m. Let u be a ground state of H. Let h ∈ C_{0}^{∞}(IR) with ∫ h(k)dk = 1 and let x_{0} ∈ IR, x_{0} ≠ 0, k_{0} ∈ IR, k_{0} ≠ 0, ω(k_{0}) = λ−inf σ(H). Choose a sequence R_{j} such that lim_{j→∞} j^{−1}R_{j} = ∞ and define h_{j} ∈ C_{0}^{∞}(IR) by setting

h_{j}(k) = j^{d/2}h(j(k − k_{0}))e^{iR_{j}k·x_{0}}.

Then ∥h_{j}∥ = 1, w − lim_{j→∞} h_{j} = 0 and lim_{j→∞}(ω(k) − ω(k_{0}))h_{j} = 0. Let

u_{j} = a^{∗}(h_{j})u.

We have lim_{j→∞} ∥u_{j}∥ = 1 and w − lim_{j→∞} u_{j} = 0. Note that u ∈ D(H^{m}) for any m, so it belongs to D(H_{0}N^{m}) for any m ∈ IR. Therefore, u_{j} ∈ D(H_{0}) ∩ D(N^{n}) ⊂ D(H) and

(H − λ)u_{j} = (H_{0} + V − λ)u_{j}
= a^{∗}(h_{j})(H − λ)u + a^{∗}(ω(k)h_{j})u + [V, a^{∗}(h_{j})]u
= a^{∗}((ω(k) − ω(k_{0}))h_{j})u + [V, a^{∗}(h_{j})]u.

It is easy to see that

∥ lim_{i→∞}(h_{i}|w_{p}∥_{L^{2}(IR^{p−1})} = 0.

Therefore, by Proposition 3.13 we get that [a^{∗}(h_{j}), V](N + 1)^{−n+1/2} → 0, when j → ∞. This implies that (H − λ)u_{j} → 0 when j → ∞, and since u_{j} tends weakly to 0, u_{j} is a Weyl sequence for λ. □

9.2 The Mourre estimate and its consequences

We denote by τ the set of thresholds

τ := σ_{pp}(H) + mIN^{∗}.

For λ ∈ IR, ϵ > 0, let I(λ, ϵ) denote [λ − ϵ, λ + ϵ]. Likewise, for a subset Θ ⊂ IR, let I(Θ, ϵ) denote the set {k ∈ IR : dist(Θ, k) ≤ ϵ}.

Theorem 9.2 Assume hypothesis (B1), or if degP = 4, hypotheses (C), (M1). Then

i) let λ ∈ IR\τ. Then there exists ϵ > 0, c_{0} > 0 and a compact operator K such that

1l_{I(λ,ϵ)}(H)[H, iA]1l_{I(λ,ϵ)}(H) ≥ c_{0}1l_{I(λ,ϵ)}(H) + K.

ii) for all [λ_{1}, λ_{2}] such that [λ_{1}, λ_{2}] ∩ τ = ∅, one has

dim1l_{[λ_{1},λ_{2}]}^{pp}(H) < ∞.

Consequently σ_{pp}(H) can accumulate only at τ, which is a closed countable set. iii) Let λ ∈ IR\(τ ∪ σ_{pp}(H)). Then there exists ϵ > 0, c_{0} > 0 such that

1l_{I(λ,ϵ)}(H)[H, iA]1l_{I(λ,ϵ)}(H) ≥ c_{0}1l_{I(λ,ϵ)}(H).

Remark 9.3 There is an example due to Simon [Si3] of a P (φ)_{2} Hamiltonian with eigenvalues embedded in [Σ + m, Σ + 2m[.

Theorem 9.4 Assume hypothesis (B2). Then the strong limiting absorption principle holds:

w − lim_{ϵ→±0}(1 + |A|)^{−r}(H − λ − iϵ)(1 + |A|)^{−r} exists

locally uniformly on σ(H)\(τ ∪ σ_{pp}(H)), for r > \frac{1}{2}. Consequently H has no singular continuous spectrum.

Thm. 9.4 is a consequence of Thm. 8.16 and the abstract Mourre theory (see [Mo], [PSS], [ABG, Thm. 7.4.1]).

Proof of Theorem 9.2. The proof will be very similar to that of [DG1, Thm. 4.3]. Let us set ω̃(k) := k · ∇ω(k) = k^{2}(k^{2} + m^{2})^{− \frac{1}{2}} . Let

d(λ) = inf_{p=1}^{∞} inf_{k_{1},...,k_{p}∈IR} { ∑_{i=1}^{p} ω̃(k_{i}) | λ − ∑_{i=0}^{p} ω(k_{i}) ∈ σ_{pp}(H)},
d̃(λ) = inf_{p=0}^{∞} inf_{k_{1},...,k_{p}∈IR} { ∑_{i=1}^{p} ω̃(k_{i}) | λ − ∑_{i=1}^{p} ω(k_{i}) ∈ σ_{pp}(H)}.

Let us note that

d̃(λ) := \{ \begin{matrix}d(λ), λ ∉ σ_{pp}(H), \\ 0, λ ∈ σ_{pp}(H). \end{matrix}

We introduce also “smeared out” versions of the functions d(λ) and d̃(λ). We set

d^{κ}(λ) := inf_{μ∈I(λ,κ)} d(μ),

= inf_{p=1}^{∞} inf_{k_{1},...,k_{p}∈IR} { ∑_{i=1}^{p} ω̃(k_{i}) | λ − ∑_{i=1}^{p} ω(k_{i}) ∈ I(σ_{pp}(H), κ)},

d̃^{κ}(λ) := inf_{μ∈I(λ,κ)} d̃(μ)

= inf_{p=0}^{∞} inf_{k_{1},...,k_{p}∈IR} { ∑_{i=1}^{p} ω̃(k_{i}) | λ − ∑_{i=1}^{p} ω(k_{i}) ∈ I(σ_{pp}(H), κ)}.

Note that the following equality holds

(9.1)  inf_{p=1}^{∞} inf_{k_{1},...,k_{p}∈IR^{p}} (d̃^{κ}(λ − ∑_{i=1}^{p} ω(k_{i})) + ∑_{i=1}^{p} ω̃(k_{i})) = d^{κ}(λ).

We will use an induction with respect to n ∈ IN. Let us first list the statements that we will show. We put E_{0} := inf σ(H). H_{1}(n) : Let ϵ > 0 and λ ∈ [E_{0}, E_{0} + nm[. Then there exists a compact operator K, an interval Δ ∋ λ such that

1l_{Δ}(H)[H, iA]1l_{Δ}(H) ≥ (d(λ) − ϵ)1l_{Δ}(H) + K.

H_{2}(n) : Let ϵ > 0 and λ ∈ [E_{0}, E_{0} + nm[. Then there exists an interval Δ ∋ λ such that

1l_{Δ}(H)[H, iA]1l_{Δ}(H) ≥ (d̃(λ) − ϵ)1l_{Δ}(H).

H_{3}(n) : Let κ > 0, ϵ_{0} > 0 and ϵ > 0. Then there exists δ > 0 such that for all λ ∈ [E_{0}, E_{0} + nm − ϵ_{0}], one has

1l_{I(λ,δ)}(H)[H, iA]1l_{I(λ,δ)}(H) ≥ (d̃^{κ}(λ) − ϵ)1l_{I(λ,δ)}(H).

S_{1}(n) : τ is a closed countable set in [E_{0}, E_{0} + nm]. S_{2}(n) : for all λ_{1} ≤ λ_{2} ≤ E_{0} + nm with [λ_{1}, λ_{2}] ∩ τ = ∅, we have dim1l_{[λ_{1},λ_{2}]}^{pp}(H) < ∞.

We will prove, for all n ∈ IN, the following implications:

H_{1}(n) ⇒ H_{2}(n),
H_{2}(n) ⇒ H_{3}(n),
H_{1}(n) ⇒ S_{2}(n),
S_{2}(n − 1) ⇒ S_{1}(n),
S_{1}(n) and H_{3}(n − 1) ⇒ H_{1}(n).

Note first that the statements H_{1}(1) and S_{1}(1) are immediate since the spectrum of H is discrete in [E_{0}, E_{0} + m[. Note also that the implication S_{2}(n − 1) ⇒ S_{1}(n) is obvious. The proof of the implications H_{1}(n) ⇒ H_{2}(n), H_{2}(n) ⇒ H_{3}(n) is a standard argument which adapt directly to the present setting (see [FH], [CFKS]). The proof of the implication H_{1}(n) ⇒ S_{2}(n) is also standard and based on the virial relation, which holds here by Thm. 8.7.

It remains to prove that S_{1}(n) and H_{3}(n − 1) ⇒ H_{1}(n).

Recall that the Hamiltonian H^{ext} acting on H ⊗ H was introduced in Subsect. 7.4. We also set A^{ext} = A ⊗ 1l + 1l ⊗ A, acting on H ⊗ H. Let us first show that for all λ ∈ [E_{0}, E_{0} + nm − ϵ_{0}[, there exists δ > 0 such that

1l_{I(λ,δ)}(H^{ext}))[H^{ext}, iA^{ext}]1l_{I(λ,δ)}(H^{ext})1l_{[1,∞[}(N_{∞})
≥ (d(λ) − \frac{2ϵ}{3})1l_{I(λ,δ)}(H^{ext})1l_{[1,∞[}(N_{∞}).

(9.2)

To simplify, let us write dΓ_{∞}(ω), dΓ_{∞}(ω̃), instead of 1l ⊗ dΓ(ω), 1l ⊗ dΓ(ω̃). We will also write B instead of B ⊗1l. Using the closedness of τ in [E_{0}, E_{0} +nm], i.e the induction hypothesis S_{1}(n), we see that

d(λ) = sup_{κ>0} d^{κ}(λ),

for λ ∈ [E_{0}, E_{0} + nm[. So we may choose κ small enough so that d^{κ}(λ) ≥ d(λ) − ϵ/3. Next using H_{3}(n − 1) we choose δ such that for λ_{1} ∈ [E_{0}, E_{0} + (n − 1)m − ϵ_{0}[, we have

1l_{I(λ_{1},δ)}(H)[H, iA]1l_{I(λ_{1},δ)}(H) ≥ (d̃^{κ}(λ_{1}) − \frac{ϵ}{3}) 1l_{I(λ_{1},δ)}(H).

Replacing λ_{1} with λ − dΓ(ω(k)), we obtain for λ ∈ [E_{0}, E_{0} + nm − ϵ_{0}[ the following estimate:

1l_{I(λ,δ)}(H + dΓ_{∞}(ω))([H, iA] + dΓ_{∞}(ω̃))1l_{I(λ,δ)}(H + dΓ_{∞}(ω))1l_{[1,∞[}(N_{∞})
≥ 1l_{I(λ,δ)}(H + dΓ_{∞}(ω)) (d̃^{κ}(λ − dΓ_{∞}(ω)) + dΓ_{∞}(ω̃) − \frac{ϵ}{3}) 1l_{[1,∞[}(N_{∞})
≥ (d^{κ}(λ) − \frac{ϵ}{3})1l_{I(λ,δ)}(H + dΓ_{∞}(ω))1l_{[1,∞[}(N_{∞})
≥ (d(λ) − \frac{2ϵ}{3})1l_{I(λ,δ)}(H + dΓ_{∞}(ω))1l_{[1,∞[}(N_{∞}),

which yields (9.2).

Now let χ ∈ C_{0}^{∞}(IR). As in the proof of Theorem 9.1

\begin{aligned}χ^{2}(H) & = I(j^{R})1l_{\{0\}}(N_{∞})I^{∗}(j^{R})χ^{2}(H) + I(j^{R})1l_{[1,∞[}(N_{∞})I^{∗}(j^{R})χ^{2}(H) \\ & = Γ(q^{R})χ^{2}(H) + I(j^{R})1l_{[1,∞[}(N_{∞})χ^{2}(H^{ext})I^{∗}(j^{R}) + o(R^{0}). \end{aligned} (9.3)

The first term of (9.3) is compact as in the proof of Thm. 9.1.

Next we use that [H, iA] equals H^{(1)} and that on D(H^{n}) H^{(1)} can be written as H_{0}^{(1)} + V^{(1)}. So on D(H^{n}) [H, iA] is similar to H with ω replaced by ω̃ and V replaced by the Wick polynomial V^{(1)}. It is then easy to see that the analog of (7.22) holds ie as an operator identity on D(H^{n}) we have: (9.4) [H^{ext}, iA^{ext}]Γ̌(j^{R}) − Γ̌(j^{R})[H, iA] ∈ (N + 1)^{n}ǒ_{N}(R^{0}),

for [H^{ext}, iA^{ext}] = [H, iA] ⊗ 1l + 1l ⊗ dΓ(ω̃). Using also Lemma 7.12, we obtain

χ(H)[H, iA]χ(H) = Γ(q^{R})χ(H)[H, iA]χ(H)
+I(j^{R})1l_{{[1,∞[}}(N_{∞})χ(H^{ext})[H^{ext}, iA^{ext}]χ(H^{ext})I^{∗}(j^{R}) + o(R^{0}),

(9.5) where the first term on the right is again compact.

Now (9.2), (9.3) and (9.5) for supp χ ⊂ [λ − δ, λ + δ], yield

χ(H)[H, iA]χ(H) ≥ (d(λ) − 2ϵ/3)χ^{2}(H) + K_{1} + o(R^{0}),

where K_{1} is compact. Picking R large enough, this proves H_{1}(n). □

10 Scattering theory of P (φ)_{2} Hamiltonians

This section is devoted to the scattering theory of P (φ)_{2} Hamiltonians. In quantum field theory, the scattering theory is usually based on the construction of the asymptotic fields, which is done in Subsect. 10.1. The unitarity of the wave operator (a result originally due to Høgh-Krohn) is shown in Subsect. 10.2, using general properties of regular CCR representations shown in Sect.

4. The asymptotic completeness property is formulated in Subsect. 10.3 and will be shown in Sect. 12.

10.1 Asymptotic fields

In all this section, we will assume the conditions (A), (Is) for s > 1.

For h ∈ 𝔥 we set h_{t} := e^{−itω(k)}h. We denote by 𝔥_{0} ⊂ 𝔥 the space C_{0}^{∞}(IR\{0}).

Theorem 10.1 i) For all h ∈ 𝔥 the strong limits

(10.1) W^{+}(h) := s- lim_{t→+∞} e^{itH}W (h_{t})e^{−itH}

exist. They are called the asymptotic Weyl operators. The asymptotic Weyl operators can be also defined using the norm limit:

W^{+}(h)(H + b)^{−n} = lim_{t→+∞} e^{itH}W (h_{t})(H + b)^{−n}e^{−itH}. (10.2)

ii) The map 𝔥 ∋ h ↦ W^{+}(h) (10.3)

is strongly continuous and for ϵ > 0, the map

𝔥 ∋ h ↦ W^{+}(h)(H + b)^{−ϵ} (10.4)

is norm continuous. iii) The operators W^{+}(h) satisfy the Weyl commutation relations:

W^{+}(h)W^{+}(g) = e^{−i \frac{1}{2} Im(h|g)}W^{+}(h + g).

iv) The Hamiltonian preserves the asymptotic Weyl operators:

e^{itH}W^{+}(h)e^{−itH} = W^{+}(h_{−t}). (10.5)

Proof. We have

W (h_{t}) = e^{−itH_{0}}W (h)e^{itH_{0}},

which implies that, as a quadratic form on D(H_{0}), one has

∂_{t}W (h_{t}) = −[H_{0}, iW (h_{t})]. (10.6)

Using (10.6) and the fact that D(H^{n}) ⊂ D(H_{0}) ∩ D(V), we have, as quadratic forms on D(H^{n}),

∂_{t}e^{itH}W (h_{t})e^{−itH} = ie^{itH}[V, W (h_{t})]e^{−itH}.

Integrating this relation we have as a quadratic form identity on D(H^{n})

(10.7) e^{itH}W (h_{t})e^{−itH} − W (h) = i ∫_{0}^{t} e^{it′H}[V, W (h_{t′})]e^{−it′H}dt′.

Using Prop. 3.13, we obtain that

[V, W (h_{t})] = W (h_{t})Ṽ_{t},

where Ṽ_{t} is the sum of Wick monomials in (3.20) with s+r ≥ 1. By stationary phase arguments, we obtain that, for h ∈ 𝔥_{0}, there exists ϵ > 0 such that

h_{t} = 1l_{{|x|≥ϵt}}h_{t} + O(t^{−∞}). (10.8)

Using then Lemma 6.3 and the form (3.21) of the symbols of Ṽ_{t}, we obtain that

∥Ṽ_{t}(N + 1)^{−n}∥ ∈ O(t^{−s}).

This shows that the identity (10.7) makes sense as an identity between bounded operators from D(H^{n}) to H. It also proves that the norm limit (10.2) exists for h ∈ 𝔥_{0}.

For h ∈ 𝔥, let h_{n} ∈ 𝔥_{0} such that h = lim_{n→∞} h_{n}. Let 0 < ϵ ≤ \frac{1}{2}. Using the first order estimates and Prop. 3.1 we obtain

\begin{aligned}∥(W (h_{n,t}) − W (h_{t}))(H + b)^{−ϵ}∥ & ≤ ∥(W (h_{n}) − W (h))(N + 1)^{−ϵ}∥∥(N + 1)^{ϵ}(H + b)^{−ϵ}∥ \\ & ≤ C(∥h_{n} − h∥^{ϵ}(∥h_{n}∥^{2} + ∥h∥)^{2} + 1), \end{aligned}

which implies

lim_{n→∞} sup_{t∈IR} ∥(W (h_{n,t}) − W (h_{t}))(H + b)^{−ϵ}∥ = 0.

This implies the existence of the norm limit (10.2) for all h ∈ 𝔥. Now (10.2) implies (10.1). This ends the proof of i). We have

\begin{aligned}∥ (W^{+}(h) − W^{+}(g)) (H + b)^{−ϵ}∥ & ≤ lim_{t→+∞} ∥e^{itH}(W (h_{t}) − W (g_{t}))(H + b)^{−ϵ}e^{−itH}∥ \\ & ≤ C(∥g − h∥^{ϵ}(∥g∥^{2} + ∥h∥)^{2} + 1), \end{aligned}

by Prop. 3.1, which implies the norm continuity of (10.4). This implies the strong continuity of (10.3) and completes the proof of ii). Finally iii) and iv) are immediate. □

It follows from the above theorem that 𝔥 ∋ h ↦ W^{+}(h) is a regular CCR representation. We next follow Sect. 2, introducing field operators, creation/annihilation operators, etc.

Theorem 10.2 i) For any h ∈ 𝔥

ϕ^{+}(h) := −i\frac{d}{ds}W^{+}(sh)_{|s=0}

defines a self-adjoint operator, called the asymptotic field, such that

W^{+}(h) = e^{iϕ^{+}(h)}.

ii) The operators ϕ^{+}(h) satisfy in the sense of quadratic forms on D(ϕ^{+}(h_{1})) ∩ D(ϕ^{+}(h_{2})) the canonical commutation relations

[ϕ^{+}(h_{2}), ϕ^{+}(h_{1})] = iIm(h_{2}|h_{1}). (10.9)

iii)

e^{itH}ϕ^{+}(h)e^{−itH} = ϕ^{+}(h_{−t}).

iv) For h_{i} ∈ 𝔥, 1 ≤ i ≤ p, D((H + i)^{p/2}) ⊂ D(Π_{1}^{p}ϕ^{+}(h_{i})), and

Π_{i=1}^{p} ϕ^{+}(h_{i})(H + i)^{−p/2} = s- lim_{t→+∞} e^{itH} Π_{i=1}^{p} ϕ(h_{i,t})e^{−itH}(H + i)^{−p/2}.

Proof. Properties i) and ii) are consequences of the fact that the asymptotic Weyl operators define a regular CCR representation (see Sect. 2). Property iii) follows from Thm. 10.1 iv). It remains to prove iv). Let us first establish the existence of the strong limit

(10.10) s- lim_{t→+∞} e^{itH}Π_{1}^{p}ϕ(h_{i,t})(H + b)^{−p/2}e^{−itH} =: R(h_{1}, …, h_{p}), for h_{i} ∈ 𝔥.

For u, v ∈ D(H^{n}), we have

\begin{matrix}\frac{∂}{∂t}(v_{t}, Π_{1}^{p}ϕ(h_{i,t})(H + b)^{−p/2}u_{t}) \\ = (v_{t}, [H, iΠ_{1}^{p}ϕ(h_{i,t})](H + b)^{−p/2}u_{t}) + (v_{t}, ∂_{t}Π_{1}^{p}ϕ(h_{i,t})(H + b)^{−p/2}u_{t}). \end{matrix}

We use again the fact that H = H_{0} + V on D(H^{n}) and the higher order estimates, which show that

[H, iΠ_{1}^{p}ϕ(h_{i,t})](H + b)^{−p/2} + ∂_{t}Π_{1}^{p}ϕ(h_{i,t})(H + b)^{−p/2}
= [V, iΠ_{1}^{p}ϕ(h_{i,t})](H + b)^{−p/2} + [H_{0}, iΠ_{1}^{p}ϕ(h_{i,t})](H + b)^{−p/2} + ∂_{t}Π_{1}^{p}ϕ(h_{i,t})(H + b)^{−p/2}
= [V, iΠ_{1}^{p}ϕ(h_{i,t})](H + b)^{−p/2},

as an identity between quadratic forms on D(H^{n}). Using then the fact that ϕ(h) maps D(N^{k}) into D(N^{k− \frac{1}{2}} ), we obtain the identity

[V, iΠ_{1}^{p}ϕ(h_{i,t})](H + b)^{−p/2} = ∑_{j=1}^{p} Π_{1}^{j−1}ϕ(h_{i,t})[V, iϕ(h_{j,t})]Π_{j+1}^{p}ϕ(h_{i,t})(H + b)^{−p/2},

as a quadratic form identity on D(H^{n}). For h ∈ 𝔥, the term [V, iϕ(h_{t})] is by Prop. 3.13 a Wick polynomial with kernels of the form w_{p}|h_{t}) or (h_{t}|w_{p}.

By a stationary phase argument, if h ∈ 𝔥_{0}, we can find ϵ_{0} > 0 such that 1l_{{|x|≤ϵ_{0}t}}h_{t} ∈ O(t^{−∞}). Using then hypothesis (Is) for s > 1, Lemma 6.3 and Prop. 3.13, we obtain

[V, iϕ(h_{t})] ∈ O_{N}(t^{−s})(N + 1)^{n}. (10.11)

Using again the higher order estimates, we obtain that if h_{i} ∈ 𝔥_{0}, 1 ≤ i ≤ p then

\frac{∂}{∂t}(v_{t}, Π_{1}^{p}ϕ(h_{i,t})(H + b)^{−p/2}u_{t}) = (v_{t}, R(t)u_{t}), u, v ∈ D(H^{n}),

where ∥R(t)(H+b)^{−n}∥ ≤ Ct^{−s}. This proves the existence of the limit (10.10) for u ∈ D(H^{n}), h_{i} ∈ 𝔥_{0}. The estimate

∥(N + 1)^{m}(ϕ(h_{1}) − ϕ(h_{2}))(N + 1)^{−m− \frac{1}{2}} ∥ ≤ C∥h_{1} − h_{2}∥

and a density argument as in the proof of Thm. 10.1 give the existence of (10.10) for u ∈ D(H^{n}), h_{i} ∈ 𝔥. Finally it follows again from the higher order estimates that Π_{1}^{p}ϕ(h_{i,t})(H +b)^{−p/2} is bounded uniformly in t, which shows the existence of (10.10) for all u ∈ H.

We prove now the identity iv) by induction on p. We have to show that D(H^{p/2}) ⊂ D(Π_{1}^{p}ϕ^{+}(h_{i})) and that R(h_{1}, …, h_{p}) = Π_{1}^{p}ϕ^{+}(h_{i})(H + b)^{−p/2}. This amounts to show that

R(h_{1}, …, h_{p}) = s- lim_{s→0}(is)^{−1}(W^{+}(sh_{1}) − 1l)Π_{2}^{p}ϕ^{+}(h_{i})(H + b)^{−p/2}.

Note that by the induction assumption D(H^{p/2}) ⊂ D(Π_{2}^{p}ϕ^{+}(h_{i})) and

(10.12) Π_{2}^{p}ϕ^{+}(h_{i})(H + b)^{−p/2} = s- lim_{t→+∞} e^{itH}Π_{2}^{p}ϕ(h_{i,t})e^{−itH}(H + b)^{−p/2}.

Using (10.12) and the fact that e^{itH}W (h_{1,t})e^{−itH} is uniformly bounded in t, we have:

(is)^{−1}(W^{+}(sh_{1}) − 1l)Π_{2}^{p}ϕ^{+}(h_{i})(H + b)^{−p/2}

= s- lim_{t→+∞} e^{itH}(is)^{−1}(W (sh_{1,t}) − 1l)Π_{2}^{p}ϕ(h_{i,t})e^{−itH}(H + b)^{−p/2}.

So to prove iv), it suffices to check that

(10.13) s- lim_{s→0} s- lim_{t→∞} e^{itH}R(s, t)e^{−itH} = 0,

for

R(s, t) = (\frac{W (sh_{1,t}) − 1l}{s} − iϕ(h_{1,t}))Π_{2}^{p}ϕ(h_{i,t})(H + b)^{−p/2}.

We recall that

sup_{|s|≤1,∥h∥≤C} ∥(\frac{W (sh) − 1l}{s})(N + 1)^{− \frac{1}{2}} ∥ < ∞, (10.14)

and

(10.15)  lim_{s→0} sup_{∥h∥≤C} ∥(\frac{W (sh) − 1l}{s} − iϕ(h))(N + 1)^{− \frac{1}{2} −ϵ}∥ = 0, ϵ > 0.

Using (10.14) and the higher order estimates, we see that R(s, t) is uniformly bounded for |s| ≤ 1, t ∈ IR, and using then (10.15) we see that lim_{s→0} sup_{t∈IR} ∥R(s, t)u∥ = 0, for u ∈ D(H^{ϵ}). This shows (10.13) and completes the proof of the theorem. □

The following theorem follows directly from Thm. 10.1 and from the properties of regular CCR representations.

Theorem 10.3 1) For any h ∈ 𝔥, the asymptotic creation and annihilation operators defined on D(a^{+♯}(h)) := D(ϕ^{+}(h)) ∩ D(ϕ^{+}(ih)) by

\begin{matrix}a^{+∗}(h) := \frac{1}{\sqrt{2}} (ϕ^{+}(h) − iϕ^{+}(ih)) , \\ a^{+}(h) := \frac{1}{\sqrt{2}} (ϕ^{+}(h) + iϕ^{+}(ih)) . \end{matrix}

are closed. ii) The operators a^{+♯} satisfy in the sense of forms on D(a^{+#}(h_{1})) ∩ D(a^{+#}(h_{2})) the canonical commutation relations

\begin{matrix}[a^{+}(h_{1}), a^{+∗}(h_{2})] = (h_{1}|h_{2})1l, \\ [a^{+}(h_{2}), a^{+}(h_{1})] = [a^{+∗}(h_{2}), a^{+∗}(h_{1})] = 0. \end{matrix}

iii) e^{itH}a^{+♯}(h)e^{−itH} = a^{+♯}(h_{−t}). (10.16)

iv) For h_{i} ∈ 𝔥, 1 ≤ i ≤ p, D((H + i)^{p/2}) ⊂ D(Π_{1}^{p}a^{+♯}(h_{i})) and

Π_{1}^{p}a^{+♯}(h_{i})(H + b)^{− \frac{p}{2}} = s- lim_{t→∞} e^{itH}Π_{1}^{p}a^{♯}(h_{i,t})(H + b)^{− \frac{p}{2}} e^{−itH}.

10.2 Asymptotic spaces and wave operators

In this subsection, we recall the construction of the asymptotic vacuum spaces and wave operators, due to Høgh-Krohn [HK]. We give a more direct proof of the unitarity of the wave operators based on the existence of a number operator for the CCR representation given by the asymptotic Weyl operators. We define the asymptotic vacuum space to be

K^{+} := {u ∈ H | a^{+}(h)u = 0, h ∈ 𝔥}.

The asymptotic space is defined as

H^{+} := K^{+} ⊗ H.

Proposition 10.4 i) K^{+} is a closed H−invariant space. ii) K^{+} is included in the domain of Π_{1}^{p}a^{+♯}(h_{i}) for h_{i} ∈ 𝔥. iii)

H_{pp}(H) ⊂ K^{+}.

Proof. i) and ii) follow by the properties of CCR relations described in Proposition 4.1. The fact that K^{+} is H−invariant follows from (10.16). To prove iii) we verify that for u ∈ D(H), Hu = λu, h ∈ 𝔥_{0}, a(h_{t})e^{−itH}u = e^{−itλ}a(h_{t})u ∈ o(1). □

The asymptotic Hamiltonian is defined by

H^{+} := K^{+} ⊗ 1l + 1l ⊗ dΓ(ω),

for

K^{+} := H|_{K^{+}}.

We also define

\begin{matrix}Ω^{+} : H^{+} → H, \\ Ω^{+}ψ ⊗ a^{∗}(h_{1}) ⋯ a^{∗}(h_{p})Ω := a^{+∗}(h_{1}) ⋯ a^{+∗}(h_{p})ψ, h_{1}, …, h_{p} ∈ 𝔥, ψ ∈ K^{+}. \end{matrix} (10.17)

The map Ω^{+} is called the wave operator. It is a particular case of the map Ω_{π} defined in Prop. 4.2. The following theorem is due to Høgh-Krohn [HK].

Theorem 10.5 Ω^{+} is a unitary map from H^{+} to H such that:

\begin{matrix}a^{+♯}(h)Ω^{+} = Ω^{+}1l ⊗ a^{♯}(h), h ∈ 𝔥, \\ HΩ^{+} = Ω^{+}H^{+}. \end{matrix}

Proof. By general properties of regular CCR representations, (see Proposition 4.2) the operator Ω^{+} is well defined and isometric. To prove that it is unitary, we will show that the CCR representation 𝔥 ∋ h ↦ W^{+}(h) admits a densely defined number operator and use Theorem 4.3.

Let n^{+} be the quadratic form associated to the CCR representation W^{+} as in Subsect. 4.2. Let us show that D(n^{+}) is dense in H. For each finite dimensional space 𝔣 ⊂ 𝔥 if

n+_{𝔣}(u) = ∑_{i=1}^{dim𝔣} ∥a^{+}(h_{i})u∥^{2},

for {h_{i}} an orthonormal base of 𝔣, we have

∥n+_{𝔣}(u)∥^{2} = lim_{t→+∞} ∑_{i=1}^{dim𝔣} ∥a(h_{i,t})e^{−itH}u∥^{2}

= lim_{t→+∞}(e^{−itH}u, dΓ(P_{𝔣,t})e^{−itH}u),

if P_{𝔣,t} is the orthogonal projection on e^{−itω}𝔣. But dΓ(P_{𝔣,t}) ≤ N, so

n+_{𝔣}(u) ≤ sup_{t} ∥N^{\frac{1}{2}} e^{−itH}u∥^{2} ≤ C∥(H + b)^{\frac{1}{2}} u∥^{2},

by the first order estimates (6.4). Therefore

D(H^{\frac{1}{2}} ) ⊂ D(n^{+}),

which implies that D(n^{+}) is dense in H and hence, by Theorem 4.3, RanΩ^{+} = H. □

10.3 Asymptotic completeness

The definition of the wave operators seems different from the one commonly used in the physics literature, where asymptotic creation operators are only applied to bound states of H, generating the so called asymptotic states. In this respect one can ask what property of the wave operators should be called asymptotic completeness. A physically important property is the fact that incoming and outgoing asymptotic vacua coincide, that is K^{+} = K^{−}, where K^{−} is defined analogously to K^{+}, with t → −∞ replacing t → +∞ in the definition of the asymptotic Weyl operators.

Since we have seen that H_{pp}(H) ⊂ K^{±}, the natural definition of asymptotic completeness is that H_{pp}(H) = K^{±}.

The following theorem is one of the main results of this paper:

Theorem 10.6 Assume hypotheses (B1),(Is) for s > 1 , or if degP = 4, hypotheses (C), (M1), (Is) for s > 1. Then the P (φ)_{2} Hamiltonian H has the asymptotic completeness property:

H_{pp}(H) = K^{±}.

Thm. 10.6 will be proved in Subsect. 12.5, as a consequence of Thm. 12.5 and of the Mourre estimate.

10.4 Extended wave operator

Recall that in Subsect. 7.5 we introduced the extended Hilbert space and the extended Hamiltonian

H^{ext} = Γ(𝔥) ⊗ Γ(𝔥), H^{ext} = H ⊗ 1l + 1l ⊗ dΓ(ω(k)).

Clearly H^{+} is a subspace of H^{ext} and

H^{+} = H^{ext}|_{H^{+}}.

Sometimes we will also need the “extended wave operator”. Its domain can be chosen to be

D(Ω^{ext,+}) := ⨁_{p=0}^{∞} D((H + b)^{\frac{p}{2}} ) ⊗ ⊗_{s}^{p}𝔥,

which is a dense subset of H^{ext}. Now we set

Ω^{ext,+} : D(Ω^{ext,+}) → H,
Ω^{ext,+}ψ ⊗ a^{∗}(h_{1}) ⋯ a^{∗}(h_{p})Ω := a^{+∗}(h_{1}) ⋯ a^{+∗}(h_{p})ψ, ψ ∈ D((H + b)^{\frac{p}{2}} ).

(10.18)

Note that Ω^{ext,+} is an unbounded operator. Clearly,

Ω^{ext,+}|_{H^{+}} = Ω^{+}. (10.19)

We will sometimes treat Ω^{+} as a partial isometry equal to zero on the orthogonal complement of H^{+} inside H^{ext}. We can then write the following identity:

Ω^{+} = Ω^{ext,+}1l_{H^{+}}. (10.20)

where 1l_{H^{+}} denotes the projection onto H^{+} inside the space H^{ext}.

10.5 Another construction of the wave operators

Recall that in Subsect. 3.9, we defined the identification operator I : Γ(𝔥) ⊗ Γ(𝔥) → Γ(𝔥).

Theorem 10.7 i) Let u ∈ D(Ω^{ext,+}). Then the limit

lim_{t→+∞} e^{itH}Ie^{−itH^{ext}}u

exists and equals Ω^{ext,+}u. ii) Let χ ∈ C_{0}^{∞}(IR). Then Ranχ(H^{ext}) ⊂ D(Ω^{ext}), Iχ(H^{ext}) and Ω^{ext,+}χ(H^{ext}) are bounded operators and

(10.21)  lim_{t→+∞} e^{itH}Ie^{−itH^{ext}}χ(H^{ext}) = Ω^{ext,+}χ(H^{ext}).

Proof. Let us first show i). Let u ∈ D((H +i)^{k/2})⊗⊗_{s}^{k}𝔥. Since by (3.6) I(H +i)^{−k/2}⊗1l_{{k}}(N_{∞}) is a bounded operator, it suffices to prove i) for u = ψ ⊗ Π_{i}^{k}a^{∗}(h_{i})Ω, ψ ∈ D((H + i)^{k/2}), h_{i} ∈ 𝔥. It follows from property (3.4) of I that

e^{itH}Ie^{−itH^{ext}}ψ ⊗ Π_{1}^{k}a^{∗}(h_{i})Ω = e^{itH}Π_{1}^{k}a^{∗}(h_{i,t})e^{−itH}ψ.

i) follows then from Thm. 10.3 iv).

To prove ii), we observe that since the boson mass is positive, vectors in H_{comp}(H^{ext}) are also in H_{comp}(H) and in H_{comp}(N_{∞}). So ii) follows from i). □

11 Propagation estimates

In this section we collect various propagation estimates about the evolution e^{−itH}, which will be used in the next section. It is essentially similar to [DG1, Sect. 6], the only difference being the control of the interaction term V, which is here much more singular. In all this section we assume hypothesis (Is) for s > 1.

We will use the following notations for various Heisenberg derivatives:

D_{0} = \frac{∂}{∂t} + [H_{0}, i·], acting on B(Γ(𝔥)),
D = \frac{∂}{∂t} + [H, i·], acting on B(H).

The following easy observation will be used to compute Heisenberg derivatives. It follows from the fact that H = H_{0} + V on D(H^{n}).

Lemma 11.1 Let IR ∋ t ↦ M(t) ∈ B(D(H), H) be of class C^{1}. Then for χ ∈ C_{0}^{∞}(IR), we have:

Dχ(H)M(t)χ(H) = χ(H)D_{0}M(t)χ(H) + χ(H)[V, iM(t)]χ(H).

We first derive a standard large velocity estimate. It means that no boson can asymptotically propagate in the region |x| > t.

Proposition 11.2 Let χ ∈ C_{0}^{∞}(IR). For R′ > R > 1, one has

∫_{1}^{∞} ∥dΓ (1l_{[R,R′]}(\frac{|x|}{t}))^{\frac{1}{2}} χ(H)e^{−itH}u∥^{2} \frac{dt}{t} ≤ C∥u∥^{2}.

Proof. Let F ∈ C^{∞}(IR) be a cutoff function equal to 1 near ∞, to 0 near the origin, with F′(s) ≥ 1l_{[R,R′]}(s). The propagation observable is Φ(t) = χ(H)dΓ (F (\frac{|x|}{t})) χ(H). The proof

is identical to that of [DG1, Prop. 6.1], except for the term χ(H)[V, idΓ(F (\frac{|x|}{t}))]χ(H), coming from the application of Lemma 11.1. By Prop. 3.13 and Lemma 6.3, [V, idΓ(F (\frac{|x|}{t}))] is a sum of Wick monomials with symbols having an L^{2} norm O(t^{−s}), s > 1, by condition (Is). Prop. 3.13 and the higher order estimates give then that χ(H)[V, idΓ(F (\frac{|x|}{t}))]χ(H) ∈ O(t^{−s}), s > 1. Thus this term is integrable in norm. □

The following proposition contains a more subtle propagation estimate. Its intuitive meaning is that along the evolution of an asymptotically free boson the instantaneous velocity ∇ω(k) and the average velocity \frac{x}{t} converge to each other as time goes to ∞.

Proposition 11.3 Let χ ∈ C_{0}^{∞}(IR), 0 < c_{0} < c_{1}. Set

Θ_{[c_{0},c_{1}]}(t) := dΓ (⟨\frac{x}{t} − ∇ω(k), 1l_{[c_{0},c_{1}]}(\frac{x}{t})(\frac{x}{t} − ∇ω(k))⟩) .

Then

∫_{1}^{∞} ∥Θ_{[c_{0},c_{1}]}(t)^{\frac{1}{2}} χ(H)e^{−itH}u∥^{2} \frac{dt}{t} ≤ C∥u∥^{2}.

Proof. The propagation observable used to prove the proposition is of the form

Φ(t) = χ(H)dΓ(b(t))χ(H),

for

b(t) := R(\frac{x}{t}) − \frac{1}{2} (⟨∇R(\frac{x}{t}), \frac{x}{t} − ∇ω(k)⟩ + hc) ,

with |∂_{x}^{α}R(x)| ≤ C_{α}, suppR ⊂ {|x| ≥ ϵ_{0} > 0}.

As above it suffices to estimate the term χ(H)[V, idΓ(b(t))]χ(H), the other terms in the Heisenberg derivative of Φ(t) being similar to those in [DG1, Prop. 6.2]. By Prop. 3.13, [V, idΓ(b(t))] is a sum of Wick momomials with symbols dΓ(b(t))w_{p,∞}, where w_{p,∞} is the kernel defined in (6.8). We use then pseudodifferential calculus, the fact that supp R ⊂ {|x| ≥ ϵ_{0}} and Lemma 6.3 to show that dΓ(b(t))w_{∞} ∈ O(t^{−s}), s > 1. By Prop. 3.13 i) this implies that χ(H)[V, idΓ(b(t))]χ(H) ∈ O(t^{−s}) and hence is integrable in norm. □

The following proposition is an improvement on Prop. 11.3.

Proposition 11.4 Let 0 < c_{0} < c_{1}, J ∈ C_{0}^{∞}({c_{0} < |x| < c_{1}}), χ ∈ C_{0}^{∞}(IR). Then

∫_{1}^{+∞} ∥dΓ (|J (\frac{x}{t}) (\frac{x}{t} − ∂ω(k)) + hc|)^{\frac{1}{2}} χ(H)e^{−itH}u∥^{2} \frac{dt}{t} < C∥u∥^{2}.

Proof. The proof is identical to [DG1, Prop. 6.3], using the argument in the proof of Prop. 11.3 to control the commutators with V. □

Note that Prop. 11.4 is still true if we replace H by H^{ext} and dΓ(b) by dΓ(b) ⊗ 1l + 1l ⊗ dΓ(b), for b = |J (\frac{x}{t}) (\frac{x}{t} − ∂ω(k)) + hc|.

The last propagation estimate of this section is the so called minimal velocity estimate, based on the Mourre estimate shown in Subsect. 9.2. Since the conjugate operator is different from the one in [DG1], we will give a more detailed proof.

Proposition 11.5 Assume condition (B1). Let χ ∈ C_{0}^{∞}(IR) be supported in IR\(τ ∪ σ_{pp}(H)). Then there exists ϵ > 0 such that

∫_{1}^{+∞} ∥Γ (1l_{[0,ϵ]} (\frac{|x|}{t})) χ(H)e^{−itH}u∥^{2} \frac{dt}{t} ≤ C∥u∥^{2}.

Proof. Let us first prove the proposition for χ supported near an energy level λ ∈ IR\τ ∪σ_{pp}(H). By Thm. 9.2, we can find χ ∈ C_{0}^{∞}(IR) equal to 1 near λ such that for some c_{0} > 0:

χ(H)[H, iA]χ(H) ≥ c_{0}χ^{2}(H). (11.1)

Let ϵ > 0 be a parameter which will be fixed later. Let q ∈ C_{0}^{∞}(|x| ≤ 2ϵ), 0 ≤ q ≤ 1, q = 1 near \{|x| ≤ ϵ\} and let q^{t} = q(\frac{x}{t}).

We use the propagation observable

Φ(t) := χ(H)Γ(q^{t})\frac{A}{t}Γ(q^{t})χ(H).

We fix cutoff functions q̃ ∈ C_{0}^{∞}(IR), χ̃ ∈ C_{0}^{∞}(IR) such that

supp q̃ ⊂ {|x| ≤ 4ϵ}, q̃q = q, χ̃χ = χ.

Let us show the following estimate:

∥N^{k} \frac{A^{m}}{t^{m}} Γ(q^{t})χ(H)∥ ≤ Cϵ^{m} + O_{ϵ}(t^{−1}), m = 1, 2. (11.2)

First note that by Lemma 3.2 iii)

A^{2m} ≤ N^{2m−1}dΓ(a^{2m}).

Next: (11.3) Γ(q^{t})dΓ(a^{2m})Γ(q^{t}) = dΓ((q^{t})^{2}, q^{t}a^{2m}q^{t}) ≤ dΓ(q^{t}a^{2m}q^{t}).

q^{t}a^{2m}q^{t} ≤ ϵ^{2m}t^{2m}ω^{2m}(k) + O(t^{2m−2})ω(k).

Therefore Γ(q^{t})dΓ(a^{2m})Γ(q^{t}) is less than

Cϵ^{2m}t^{2m}dΓ(ω^{2m}) + Ct^{2m−2}dΓ(ω) ≤ Cϵ^{2m}t^{2m}dΓ(ω)^{2m} + Ct^{2m−2}dΓ(ω).

Therefore

∥N^{k} \frac{A^{m}}{t^{m}} Γ(q^{t})χ(H)u∥^{2} ≤ Cϵ^{2m}∥N^{k+m− \frac{1}{2}} H_{0}^{m}χ(H)u∥^{2} + Ct^{−2}∥N^{k+m− \frac{1}{2}} H_{0}χ(H)u∥^{2}.

Then we apply the high order estimates.

Now (11.2) implies the uniform boundedness of Φ(t).

Let us compute the Heisenberg derivative of Φ(t). Using Lemma 11.1, we have, for d_{0}q^{t} = ∂_{t}q^{t} + [ω, iq^{t}]:

DΦ(t) = χ(H)dΓ(q^{t}, d_{0}q^{t})\frac{A}{t}Γ(q^{t})χ(H) + hc
+χ(H)[V, iΓ(q^{t})]\frac{A}{t}Γ(q^{t})χ(H) + hc
+t^{−1}χ(H)Γ(q^{t})[H, iA]Γ(q^{t})χ(H)
−t^{−1}χ(H)Γ(q^{t})\frac{A}{t}Γ(q^{t})χ(H)
=: R_{1}(t) + R_{2}(t) + R_{3}(t) + R_{4}(t).

(11.4)

We have used the fact that Γ(q^{t}) preserves D(H_{0}) and D(N^{n}) to expand the commutator [H, iΦ(t)] in (11.4).

Let us first estimate R_{2}(t). By Lemma 3.17 and Lemma 6.3,

[V, iΓ(q^{t})] ∈ (N + 1)^{−n}O_{N}(t^{−s}), s > 1,

Therefore by (11.2) R_{2}(t) ∈ O(t^{−s}), s > 1. (11.5)

We consider next R_{1}(t). We have:

d_{0}q^{t} = −\frac{1}{2t}⟨\frac{x}{t} − ∇ω(k), ∇q(\frac{x}{t})⟩ + hc + r^{t} =: \frac{1}{t}g^{t} + r^{t},

where r^{t} ∈ O(t^{−2}). By the higher order estimates (7.1) ∥χ(H)dΓ(q^{t}, r^{t})∥ ∈ O(t^{−2}), which using (11.2) yields

∥χ(H)dΓ(q^{t}, r^{t})\frac{A}{t}Γ(q^{t})χ(H)∥ ∈ O(t^{−2}).

Then we set

B_{1} := χ(H)dΓ(q^{t}, g^{t})(N + 1)^{− \frac{1}{2}} , B_{2}^{∗} := (N + 1)^{\frac{1}{2}} \frac{A}{t}Γ(q^{t})χ(H),

and use the inequality

\begin{matrix}χ(H)dΓ(q^{t}, g^{t})\frac{A}{t}Γ(q^{t})χ(H) + hc & = t^{−1}B_{1}B_{2}^{∗} + t^{−1}B_{2}B_{1}^{∗} \\ ≥ −t^{−1}B_{1}B_{1}^{∗} − t^{−1}B_{2}B_{2}^{∗}. \end{matrix} (11.6)

We have

\begin{aligned}−B_{2}B_{2}^{∗} & = −χ(H)Γ(q^{t})\frac{A^{2}}{t^{2}} (N + 1)Γ(q^{t})χ(H) \\ & = χ(H)Γ(q^{t})χ̃(H)Γ(q̃^{t})\frac{A^{2}}{t^{2}} (N + 1)Γ(q̃^{t})χ̃(H)Γ(q^{t})χ(H) + O(t^{−1}) \\ & ≥ −ϵ^{2}C_{1}χ(H)Γ^{2}(q^{t})χ(H) + O(t^{−1}), \end{aligned} (11.7)

where we used Lemma 7.11 and the boundedness of \frac{A^{2}}{t^{2}} (N + 1)Γ(q^{t})χ(H) in the first step and the estimate analogous to (11.2):

χ̃(H)Γ(q̃^{t})\frac{A^{2}}{t^{2}} (N + 1)Γ(q̃^{t})χ̃(H) ≤ C_{1}ϵ^{2} + O(t^{−2})

in the second step. Next we use that by Lemma 3.4 v):

∥(N + 1)^{− \frac{1}{2}} dΓ(q^{t}, g^{t})u∥ ≤ ∥dΓ(g^{t∗}g^{t})^{\frac{1}{2}} u∥, u ∈ H,

to obtain:

\begin{aligned}|(u, B_{1}^{∗}B_{1}u)| & = ∥(N + 1)^{− \frac{1}{2}} dΓ(q^{t}, g^{t})χ(H)u∥^{2} \\ & ≤ ∥dΓ(g^{t∗}g^{t})^{\frac{1}{2}} χ(H)u∥^{2}, u ∈ H. \end{aligned}

Using Prop. 11.3, we obtain

∫_{1}^{+∞} ∥B_{1}e^{−itH}u∥^{2} \frac{dt}{t} ≤ C∥u∥^{2}. (11.8)

Next we use Lemma 7.11, to write:

\begin{aligned}R_{3}(t) & = t^{−1}Γ(q^{t})χ(H)[H, iA]χ(H)Γ(q^{t}) + O(t^{−2}) \\ & ≥ C_{0}t^{−1}Γ(q^{t})χ^{2}(H)Γ(q^{t}) − Ct^{−2} \\ & ≥ C_{0}t^{−1}χ(H)Γ^{2}(q^{t})χ(H) − Ct^{−2}. \end{aligned} (11.9)

It remains to estimate R_{4}(t). We have

\begin{aligned}R_{4}(t) & = −t^{−1}χ(H)Γ(q^{t})\frac{A}{t}Γ(q^{t})χ(H) \\ & = −t^{−1}χ(H)Γ(q^{t})χ̃(H)Γ(q̃^{t})\frac{A}{t}Γ(q̃^{t})χ̃(H)Γ(q^{t})χ(H) \\ & ≥ −ϵC_{2}t^{−1}χ(H)Γ(q^{t})^{2}χ(H) + O(t^{−2}). \end{aligned} (11.10)

Collecting (11.7), (11.9) and (11.10), we obtain

\begin{matrix}−ϵ^{2}t^{−1}B_{2}^{∗}(t)B_{2}(t) + R_{3}(t) + R_{4}(t) \\ ≥ (−ϵ^{2}C_{1} + C_{0} − ϵC_{2})t^{−1}χ(H)Γ(q^{t})^{2}χ(H) − Ct^{−2}. \end{matrix} (11.11)

We pick now ϵ small enough so that C̃_{0} = −ϵ^{2}C_{1} + C_{0} − ϵC_{2} > 0. Using (11.5), (11.8) and (11.11) we conclude that

DΦ(t) ≥ \frac{C̃_{0}}{t} χ(H)Γ^{2}(q^{t})χ(H) − R(t) − Ct^{−s}, s > 1.

where R(t) is integrable along the evolution. By the standard argument, this proves the proposition for χ with support close enough to an energy level λ ⊂ IR\(τ ∪ σ_{pp}(H)). To prove the proposition for all χ supported in IR\(τ ∪ σ_{pp}(H)) we argue as in [DG2, Prop. 4.4.7]. □

12 Asymptotic completeness

12.1 Existence of asymptotic localizations

Theorem 12.1 Assume hypothesis (Is), s > 1. Let q ∈ C_{0}^{∞}(IR), 0 ≤ q ≤ 1, q = 1 on a neighborhood of zero. Set q^{t}(x) = q(\frac{x}{t}). Then there exists

(12.1) s- lim_{t→∞} e^{itH}Γ(q^{t})e^{−itH} =: Γ^{+}(q).

We have Γ^{+}(qq̃) = Γ^{+}(q)Γ^{+}(q̃), (12.2)

0 ≤ Γ^{+}(q) ≤ Γ^{+}(q̃) ≤ 1l, if 0 ≤ q ≤ q̃ ≤ 1, (12.3)

[H, Γ^{+}(q)] = 0. (12.4)

Proof. Let us first prove the existence of (12.1). Using Lemma 7.11 and a density argument, it suffices to prove the existence of

s- lim_{t→+∞} e^{itH}χ(H)Γ(q^{t})χ(H)e^{−itH}

for χ ∈ C_{0}^{∞}(IR). We compute the Heisenberg derivative:

χ(H)DΓ(q^{t})χ(H) = χ(H)dΓ(q^{t}, d_{0}q^{t})χ(H) + χ(H)[V, iΓ(q^{t})]χ(H),

by Lemma 11.1. From Lemma 3.17, Lemma 6.3 and hypothesis (Is) we obtain

∥χ(H)[V, iΓ(q^{t})]χ(H)∥ ∈ O(t^{−s}). (12.5)

Next we compute:

d_{0}q^{t} = \frac{1}{t}g^{t} + r^{t},

where

g^{t} = −\frac{1}{2} ((\frac{x}{t} − ∂ω(k))∂q(\frac{x}{t}) + hc)

and r^{t} ∈ O(t^{−2}). Using Lemma 3.4 v) and the higher order estimates, we obtain that

∥χ(H)dΓ(q^{t}, r^{t})χ(H)∥ ∈ O(t^{−2}). (12.6)

On the other hand by Lemma 3.4 v) we have

|(u|χ(H)dΓ(q^{t}, g^{t})χ(H)u)| ≤ ∥dΓ(|g^{t}|)^{\frac{1}{2}} χ(H)u∥^{2}. (12.7)

Hence the existence of the limit (12.1) follows from (12.5)–(12.7), Proposition 11.4 and Lemma A.1.

(12.4) follows by Lemma 7.11. (12.2) follows from

Γ(q^{t}q̃^{t}) = Γ(q^{t})Γ(q̃^{t}).

□

An analogous theorem is true for the free Hamiltonian, but it is much easier. It follows within each n-particle sector by the stationary phase method. Note that in the free case one does not need to assume that the cutoff function q is one at zero.

Proposition 12.2 Let q ∈ C_{0}^{∞}(IR), 0 ≤≤ 1. Then

s- lim_{t→∞} e^{itdΓ(ω)}Γ(q^{t})e^{−itdΓ(ω)} = Γ(q(∇ω)).

12.2 Projection P_{0}^{+}

Theorem 12.3 Let {q_{n}} ∈ C_{0}^{∞}(IR) be a decreasing sequence of functions such that 0 ≤ q_{n} ≤ 1, q = 1 on a neighborhood of 0 and ∩_{n=1}^{∞}supp q_{n} = {0}. Then

(12.8) P_{0}^{+} := s- lim_{n→∞} Γ^{+}(q_{n}) exists.

P_{0}^{+} does not depend on the choice of the sequence {q_{n}}. It is an orthogonal projection satisfying

[H, P_{0}^{+}] = 0.

Besides RanP_{0}^{+} ⊂ K^{+}. (12.9)

The range of P_{0}^{+} can be interpreted as the space of states asymptotically containing no bosons away from the origin. Proof. The existence of P_{0}^{+} and the fact that it is a projection follow from (12.2), (12.3) and Lemma A.3. To show that P_{0}^{+} does not depend on the choice of {q_{n}}, we pick two sequences {q_{n}}, {q̃_{n}}. There exist for each n ∈ IN an index m_{n} such that q_{n} ≥ q̃_{m_{n}}, q̃_{n} ≥ q_{m_{n}}. Hence by (12.3) we see that

s- lim_{n→∞} Γ^{+}(q_{n}) = s- lim_{n→∞} Γ^{+}(q̃_{n}).

The fact that [H, P_{0}^{+}] = 0 follows from (12.4).

Let us now show (12.9). We know that RanP_{0}^{+} is invariant wrt H, hence D(H) ∩ RanP_{0}^{+} is dense in RanP_{0}^{+}. Besides, K^{+} is closed. Thus it is enough to show that D(H) ∩ RanP_{0}^{+} ⊂ K^{+}.

Let u ∈ D(H) ∩ RanP_{0}^{+}. We are going to show that

(H + b)^{− \frac{1}{2}} a^{+}(h)u = 0, h ∈ 𝔥, (12.10)

which will imply u ∈ K^{+}. By the continuity of 𝔥 ∋ h ↦ (H + b)^{− \frac{1}{2}} a^{+}(h) it is enough to assume that h ∈ 𝔥_{0}. By stationary phase arguments we may choose q ∈ C_{∞}(IR) with 0 ≤ q ≤ 1, q(0) = 1 and supp q contained in a sufficiently small neighborhood of 0 so that q^{t}h_{t} ∈ o(1). Then

u = lim_{t→∞} e^{itH}Γ(q^{t})e^{−itH}u,

(H + b)^{− \frac{1}{2}} a^{+}(h) = s- lim_{t→∞} e^{itH}(H + b)^{− \frac{1}{2}} a(h_{t})e^{−itH}.

Hence

(H + b)^{− \frac{1}{2}} a^{+}(h)u = lim_{t→∞} e^{itH}(H + b)^{− \frac{1}{2}} a(h_{t})Γ(q^{t})(H + b)^{− \frac{1}{2}} e^{−itH}(H + b)^{\frac{1}{2}} u
= lim_{t→∞} e^{itH}(H + b)^{− \frac{1}{2}} Γ(q^{t})a(q^{t}h_{t})(H + b)^{− \frac{1}{2}} e^{−itH}(H + b)^{\frac{1}{2}} u.

(12.11)

But since q^{t}h_{t} ∈ o(1), a(q^{t}h_{t})(H + b)^{− \frac{1}{2}} ∈ o(1) and therefore (12.11) vanishes. □

12.3 Geometric inverse wave operators

Let j_{0} ∈ C_{0}^{∞}(IR), j_{∞} ∈ C^{∞}(IR), 0 ≤ j_{0}, 0 ≤ j_{∞}, j_{0}^{2} + j_{∞}^{2} ≤ 1, j_{0} = 1 near 0 (and hence j_{∞} = 0 near 0). Set j := (j_{0}, j_{∞}). Set also j^{t} = (j_{0}^{t}, j_{∞}^{t}), where j_{0}^{t}(x) = j_{0}(\frac{x}{t}), j_{∞}^{t}(x) = j_{∞}(\frac{x}{t}).

As in Subsect. 3.10, we introduce the operator I(j^{t}) : Γ(𝔥) ⊗ Γ(𝔥) → Γ(𝔥).

Theorem 12.4 Assume hypothesis (Is) for s > 1.

i) The following limits exist:

(12.12) s- lim_{t→+∞} e^{itH^{ext}}I^{∗}(j^{t})e^{−itH},

(12.13) s- lim_{t→+∞} e^{itH}I(j^{t})e^{−itH^{ext}}.

If we denote (12.12) by W^{+}(j), then (12.13) equals W^{+}(j)^{∗}. ii) For a bounded Borel function F one has

W^{+}(j)F (H) = F (H^{ext})W^{+}(j).

iii) Let q_{0}, q_{∞} ∈ C^{∞}(IR), ∇q_{0}, ∇q_{∞} ∈ C_{0}^{∞}(IR), 0 ≤ q_{0}, q_{∞} ≤ 1, q_{0} = 1 near 0. Set j̃ := (j̃_{0}, j̃_{∞}) := (q_{0}j_{0}, q_{∞}j_{∞}). Then

Γ^{+}(q_{0}) ⊗ Γ(q_{∞}(∇ω))W^{+}(j) = W^{+}(j̃).

iv) Let q ∈ C^{∞}(IR), ∇q ∈ C_{0}^{∞}(IR), 0 ≤ q ≤ 1, q = 1 near 0. Then

W^{+}(j)Γ^{+}(q) = W^{+}(qj),

where qj = (qj_{0}, qj_{∞}).

v) Let j̃ = (j̃_{0}, j̃_{∞}) be another pair satisfying the conditions stated at the beginning of this subsection. (Note that j̃_{0}j_{0} + j̃_{∞}j_{∞} ≤ 1 and j̃_{0}j_{0} = 1 near zero). Then

W^{+}(j̃)^{∗}W^{+}(j) = Γ^{+}(j̃_{0}j_{0} + j̃_{∞}j_{∞}),

In particular, if j_{0}^{2} + j_{∞}^{2} = 1, then W^{+}(j) is isometric. vi) Let j_{0} + j_{∞} = 1. If χ ∈ C_{0}^{∞}(IR), then

Ω^{ext,+}χ(H^{ext})W^{+}(j) = χ(H).

Proof. Let us first prove the existence of the limit (12.12), the case of (12.13) being similar. Using Lemma 7.12 and a density argument, it suffices to prove the existence of

s- lim_{t→∞} e^{itH^{ext}}χ(H^{ext})I^{∗}(j^{t})χ(H)e^{−itH},

for some χ ∈ C_{0}^{∞}(IR). We compute the asymmetric Heisenberg derivative

χ(H^{ext})ĎI^{∗}(j^{t})χ(H) = χ(H^{ext})Ď_{0}I^{∗}(j^{t})χ(H)
+iχ(H^{ext})(V ⊗ 1lI^{∗}(j^{t}) − I^{∗}(j^{t})V)χ(H).

From Lemma 3.17, Lemma 6.3 and hypothesis (Is), we obtain ∥χ(H^{ext})(V ⊗ 1lI^{∗}(j^{t}) − I^{∗}(j^{t})V)χ(H)∥ ∈ O(t^{−s}). (12.14) On the other hand by Lemma 3.11, we have Ď_{0}I^{∗}(j^{t}) = dI^{∗}(j^{t}, ď_{0}j^{t}), and, by pseudodifferential calculus,

ď_{0}^{t}j^{t} = \frac{1}{t}k^{t} + r^{t},

where

k^{t} = (k_{0}^{t}, k_{∞}^{t}), k_{ϵ}^{t} = −\frac{1}{2}((\frac{x}{t} − ∂ω(k))∂j_{ϵ}(\frac{x}{t}) + hc), ϵ = 0, ∞

and r^{t} ∈ O(t^{−2}). Using Lemma 3.11 v) and the higher order estimates we obtain ∥χ(H^{ext})dI^{∗}(j^{t}, r^{t})χ(H)∥ ∈ O(t^{−2}). (12.15) Using then Lemma 3.11 iv), we obtain

|(u_{2}|χ(H^{ext})dI^{∗}(j^{t}, k^{t})χ(H)u_{1})|

≤ ∥(dΓ(|k_{0}^{t}|)^{\frac{1}{2}} ⊗ 1l)χ(H^{ext})u_{2}∥∥dΓ(|k_{0}^{t}|)^{\frac{1}{2}} χ(H)u_{1}∥
+∥(1l ⊗ dΓ(|k_{∞}^{t}|)^{\frac{1}{2}} )χ(H^{ext})u_{2}∥∥dΓ(|k_{∞}^{t}|)^{\frac{1}{2}} χ(H)u_{1}∥.

(12.16)

Hence the existence of the limit (12.12) follows from (12.14)–(12.16), Proposition 11.4 and Lemma A.2.

ii) follows from Lemma 7.12. iii) follows from Prop. 12.2 and the fact that

Γ(q_{0}^{t}) ⊗ Γ(q_{∞}^{t})I^{∗}(j^{t}) = I^{∗}(j̃^{t}).

iv) follows from

I^{∗}(j^{t})Γ(q^{t}) = I^{∗}((jq)^{t}).

v) follows from

I(j̃^{t})I^{∗}(j^{t}) = Γ(j̃_{0}^{t}j_{0}^{t} + j̃_{∞}^{t}j_{∞}^{t}).

Up to technical details due to the unboundedness of I, vi) can be considered as a special case of v) with j̃ = (1, 1) . To prove vi) we note that

H^{ext}1l_{[k,∞[}(N_{∞}) ≥ mk + E_{0},

where E_{0} = inf σ(H). Hence for χ ∈ C_{0}^{∞}(IR) we can find n ∈ IN such that χ(H^{ext})1l_{]n,∞[}(N_{∞}) = 0. (12.17) Therefore

Ω^{ext,+}χ(H^{ext})W^{+}(j) = Ω^{ext,+}1l_{[0,n]}(N_{∞})χ(H^{ext})W^{+}(j)  (1)
= s- lim_{t→∞} e^{itH}I1l_{[0,n]}(N_{∞})χ(H^{ext})I^{∗}(j^{t})e^{−itH} (2)
= s- lim_{t→∞} e^{itH}I1l_{[0,n]}(N_{∞})I^{∗}(j^{t})e^{−itH}χ(H)  (3),

(12.18) using (12.17) in step (1), Thm. 10.7 ii) and Thm. 12.4 i) in step (2) and Lemma 7.12 and the boundedness of I1l_{[0,n]}(N_{∞})(N_{0})^{−n} in step (3).

Next we claim that

∥I1l_{]n,∞[}(N_{∞})I^{∗}(j^{t})(N + 1)^{−1}∥ ≤ C(n + 1)^{−1}. (12.19)

In fact the operator

I1l_{]n,∞[}(N_{∞})I^{∗}(j^{t}) = Γ(i)1l_{]n,∞[}(dΓ( ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ))Γ(j^{t∗})

commutes with N. On ⊗_{s}^{n}𝔥 it can be written as

∑_{♯{i|ϵ_{i}=∞}>n} j_{ϵ_{1}}^{t} ⊗ ⋯ ⊗ j_{ϵ_{n}}^{t},

where the indices ϵ_{i} take the values 0, ∞. This explicit expression and the fact that j_{0} + j_{∞} = 1 imply

\begin{matrix}∥I1l_{]n,∞[}(N_{∞})I^{∗}(j^{t})∥ ≤ 1, \\ I1l_{]n,∞[}(N_{∞})I^{∗}(j^{t})1l_{[0,n]}(N) = 0, \end{matrix}

which yields (12.19). Hence

lim_{n→∞} lim sup_{t→∞} ∥e^{itH}I1l_{]n,∞[}(N_{∞})I^{∗}(j^{t})e^{−itH}χ(H)∥ = 0.

Since n can be chosen arbitrarily big and

II^{∗}(j^{t}) = 1l,

(12.18) equals χ(H). □

12.4 Geometric asymptotic completeness

In this subsection we will show that

RanP_{0}^{+} = K^{+}.

We call this property geometric asymptotic completeness. It will be convenient to work in the extended space H^{ext} and to treat Ω^{+} as a partial isometry Ω^{+} : H^{ext} → H, as explained in Subsect. 10.4.

We will give an explicit construction of the inverse wave operator Ω^{+∗} in terms of the geometric inverse wave operators W^{+}(j).

Theorem 12.5 Assume hypothesis (Is) for s > 1. Let j_{n} = (j_{0,n}, j_{∞,n}) satisfy the conditions of Subsect. 12.3. Additionally, assume that j_{0,n} + j_{∞,n} = 1 and that for any ϵ > 0, there exists m such that, for n > m, supp j_{0,n} ⊂ [−ϵ, ϵ] Then

Ω^{+∗} = w − lim_{n→∞} W^{+}(j_{n}).

Besides

K^{+} = RanP_{0}^{+}.

Proof. Let q ∈ C_{0}^{∞}(IR), q = 1 in a neighborhood of 0, 0 ≤ q ≤ 1. For sufficiently big n we have qj_{0,n} = j_{0,n}. Therefore, for sufficiently big n by Thm. 12.4 iii)

(Γ^{+}(q) ⊗ 1l)W^{+}(j_{n}) − W^{+}(j_{n}) = 0.

Hence

w − lim_{n→∞} (P_{0}^{+} ⊗ 1lW^{+}(j_{n}) − W^{+}(j_{n})) = 0. (12.20)

Let now u ∈ H, χ ∈ C_{0}^{∞}(IR). We have

Ω^{+∗}χ(H) = Ω^{+∗}Ω^{ext,+}χ(H^{ext})W^{+}(j_{n})  (1)
= w − lim_{n→∞} Ω^{+∗}Ω^{ext,+}χ(H^{ext})W^{+}(j_{n})  (2)
= w − lim_{n→∞} Ω^{+∗}Ω^{ext,+}χ(H^{ext})P_{0}^{+} ⊗ 1lW^{+}(j_{n}) (3)
= w − lim_{n→∞} P_{0}^{+} ⊗ 1lχ(H^{ext})W^{+}(j_{n})  (4)
= w − lim_{n→∞} P_{0}^{+} ⊗ 1lW^{+}(j_{n})χ(H)  (5)
= w − lim_{n→∞} W^{+}(j_{n})χ(H)  (6).

We used Theorem 12.4 vi) in step (1); step (2) is obvious – we just added w − lim_{n→∞} to a constant sequence; (12.20) was used in step (3) (note that P_{0}^{+} ⊗ 1l commutes with χ(H^{ext})); in step (4) we used K^{+} ⊃ RanP_{0}^{+}, Ω^{ext,+}1l_{K^{+}} ⊗ 1l = Ω^{+}, Ω^{+∗}Ω^{+} = 1l_{K^{+}} ⊗ 1l; in step (5) we used Theorem 12.4 ii); finally in step (6) we used again (12.20). The arbitrariness of χ ∈ C_{0}^{∞}(IR) and a density argument imply

Ω^{+∗} = w − lim_{n→∞} W^{+}(j_{n}).

Therefore by (12.20), (P_{0}^{+} ⊗ 1l)Ω^{+∗} = Ω^{+∗}, ie

RanΩ^{+∗} ⊂ RanP_{0}^{+} ⊗ Γ(𝔥) ⊂ K^{+} ⊗ Γ(𝔥).

But by construction

RanΩ^{+∗} = K^{+} ⊗ Γ(𝔥).

Hence K^{+} ⊗ Γ(𝔥) = RanP_{0}^{+} ⊗ Γ(𝔥), and therefore

K^{+} = RanP_{0}^{+}.

□

12.5 Asymptotic completeness

In this subsection, we will prove Thm. 10.6.

Proof of Thm. 10.6. By Proposition 10.4 and geometric asymptotic completeness we already know that

H_{pp}(H) ⊂ K^{+} = RanP_{0}^{+}.

It remains to prove that P_{0}^{+} ≤ 1l_{pp}(H). Let χ ∈ C_{0}^{∞}(IR\(τ ∪ σ_{pp}(H))). We deduce from Prop. 11.5 in Sect. 11 that there exists ϵ > 0 such that for q ∈ C_{0}^{∞}([−ϵ, ϵ]) with q(x) = 1 for |x| < ϵ/2 we have

∫_{1}^{+∞} ∥Γ(q^{t})χ(H)e^{−itH}u∥^{2} \frac{dt}{t} ≤ c∥u∥^{2}.

Since ∥Γ(q^{t})χ(H)e^{−itH}u∥ → ∥Γ^{+}(q)χ(H)u∥, we have Γ^{+}(q)χ(H) = 0. This implies that

P_{0}^{+} ≤ 1l_{τ∪σ_{pp}}(H).

Since τ is a closed countable set and σ_{pp}(H) can accumulate only at τ, we see that 1l_{pp}(H) = 1l_{τ∪σ_{pp}}(H). This completes the proof of the theorem. □

A Appendix

The following lemma describes an argument commonly used to prove the so called propagation estimates (see [DG1, Sect. 8.4] and references therein).

Lemma A.1 Let H be a self-adjoint operator and D the corresponding Heisenberg derivative

D := \frac{d}{dt} + i[H, ·].

Suppose that Φ(t) is a uniformly bounded family of self-adjoint operators. Suppose that there exist C_{0} > 0 and operator valued functions B(t) and B_{i}(t), i = 1, …, n, such that

\begin{matrix}DΦ(t) ≥ C_{0}B^{∗}(t)B(t) − ∑_{i=1}^{n} B_{i}^{∗}(t)B_{i}(t), \\ ∫_{1}^{∞}∥B_{i}(t)e^{−itH}ϕ∥^{2}dt ≤ C∥ϕ∥^{2}, i = 1, …, n. \end{matrix}

Then there exists C_{1} such that

∫_{1}^{∞} ∥B(t)e^{−itH}ϕ∥^{2}dt ≤ C_{1}∥ϕ∥^{2}. (A.1)

Next we describe how one uses propagation estimates to prove the existence of asymptotic observables.

Lemma A.2 Let H_{1} and H_{2} be two self-adjoint operators. Let _{2}D_{1} be the corresponding asymmetric Heisenberg derivative:

2D_{1}Φ(t) := \frac{d}{dt}Φ(t) + iH_{2}Φ(t) − iΦ(t)H_{1}.

Suppose that Φ(t) is a uniformly bounded function with values in self-adjoint operators. Let D_{1} ⊂ H be a dense subspace. Assume that

|(ψ_{2}|_{2}D_{1}Φ(t)ψ_{1})| ≤ ∑_{i=1}^{n} ∥B_{2i}(t)ψ_{2}∥∥B_{1i}(t)ψ_{1}∥,

∫_{1}^{∞}∥B_{2i}(t)e^{−itH_{2}}ϕ∥^{2}dt ≤ C∥ϕ∥^{2}, ϕ ∈ H, i = 1, …, n,

∫_{1}^{∞}∥B_{1i}(t)e^{−itH_{1}}ϕ∥^{2}dt ≤ C∥ϕ∥^{2}, ϕ ∈ D_{1}, i = 1, …, n.

Then the limit

s- lim_{t→∞} e^{itH_{2}}Φ(t)e^{−itH_{1}}

exists.

The proof of the following lemma is given in [DG1]:

Lemma A.3 Let Q_{n} be a commuting sequence of selfadjoint operators such that:

0 ≤ Q_{n} ≤ 1, Q_{n+1} ≤ Q_{n}, Q_{n+1}Q_{n} = Q_{n+1}.

Then the limit

Q = s- lim_{n→∞} Q_{n}.

exists and is a projection.

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