MaPhySto
Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

MPS-RR 1999-18
May 1999

# Phillips-Sarnak's Conjecture for Hecke Groups with Primitive Character

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## Abstract

We prove a conjecture of Phillips and Sarnak about the disappearance of embedded eigenvalues for the Laplacian $A(\Gamma_{0}(N),\chi)$, where $\chi$ is a primitive character mod $N$, under an analytic family of character perturbations $\chi^{(\alpha)}$ with $\chi^{(0)}=\chi$. Eigenvalues with odd eigenfunctions turn into resonances and odd eigenfunctions into residues of Eisenstein series. This indicates that the Weyl law is violated by the operators $A(\Gamma_{0}(N),\chi^{(\alpha)})$ for $0<| \alpha| <\varepsilon$ and some $\varepsilon>0$.

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