- Workshop on Mathematical Physics
[MaPhySto logo]
The Danish National Research Foundation:
Network in Mathematical Physics and Stochastics

Funded by The Danish National Research Foundation

Workshop on Mathematical Physics

Friday October 1, 2004, from 13-??
Department of Mathematical Sciences, University of Aarhus
Auditorium D4, Building 531


Coupled Harmonic Oscillators in the Thermodynamic Limit
Jacob Schach Møller (Aarhus)
In this talk we study the low-lying excited states of coupled harmonic oscillators. We verify that the wellknown perturbative and semiclassical pictures persist in the thermodynamic limit. The two key ingredients are: A detailed analysis of the logarithm of the ground state eigenfunction. The fact that the Schrödinger operator is (up to a constant) a zeroth order Hodge Laplacian, associated with a twisted exterior derivative introduced by Witten.
Stark hamiltonians: dynamics of 1D Bloch electrons in high momentum regime and growth of energy for driven quantum rings.
Gheorghe Nenciu (Bucharest)
We consider the dynamics of a 1D Bloch electron subjected to a constant electric field $E$, described in the temporal gauge by the Hamiltonian
 H}^{SW}(t)=(p-eEt)^2+V_{{\rm per }}$$
The periodic potential is supposed to be less singular than the $\delta$-like potential (Dirac comb). The main result is a rigor ous proof of Ao's claim that for a large class of initial conditions (high momentum regime) there is no dynamical localization. The proof is ba sed on the mathematical substantiation of the two simplifying assumptions made in physical literature: the transitions between far away bands can be neglected and the transitions at the quasi-crossing can be describe d by Landau-Zener like formulae. By Avron and Nemirovski connection our results implies also the increase of energy for weakly singular driven quantum rings.
The multiconfiguration methods for atoms and molecules
Mathieu Lewin (Copenhagen)
We present our recent results on the multiconfiguration methods, which are used in Quantum Chemistry for the description of non-relativistic electrons in atoms and molecules. They are the natural generalization of the well-known Hartree-Fock theory. In particular, we present a new definition of approximate excited states which are certain critical points of the multiconfiguration functional. We also show how this definition can be used in practice to compute the first excited state and present numerical results.
On a Bogoliubov-Dirac-Fock theory
Christian Hainzl (Copenhagen)
We study a generalized Dirac-Fock theory where we take not only real electrons, but also the electrons filling the Dirac sea. Assuming that the uniformly filled sea has no physical consequence we derive a functional which is bounded from below. This functional we minimize and get as absolut minimizer a deformed vacuum in the presence of external fields. In the case of atoms, we minimize in fixed charge sector. As a consequence we derive a selfconsistent equation for the minimizer which consists of the know Dirac-Fock mean field operator on the one hand and plus polarization corrections on the other hand. The crucial point is that we derive these equations simply by minimizing.