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DISSOLVING OF CUSP FORMS: HIGHER-ORDER FERMI’S GOLDEN RULES

Part of: Partial differential equations on manifolds; differential operators Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  21 January 2013

Yiannis N. Petridis  and
Morten S. Risager
Yiannis N. Petridis
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, U.K. (email: i.petridis@ucl.ac.uk)
Morten S. Risager
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark (email: risager@math.ku.dk)
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Abstract

For a hyperbolic surface, embedded eigenvalues of the Laplace operator are unstable and tend to dissolve into scattering poles i.e. become resonances. A sufficient dissolving condition was identified by Phillips–Sarnak and is elegantly expressed in Fermi’s golden rule. We prove formulas for higher approximations and obtain necessary and sufficient conditions for dissolving a cusp form with eigenfunction $u_j$ into a resonance. In the framework of perturbations in character varieties, we relate the result to the special values of the $L$-series $L(u_j\otimes F^n, s)$. This is the Rankin–Selberg convolution of $u_j$ with $F(z)^n$, where $F(z)$is the antiderivative of a weight two cusp form. In an example we show that the above-mentioned conditions force the embedded eigenvalue to become a resonance in a punctured neighborhood of the deformation space.

MSC classification

Secondary: 11F72: Spectral theory; Selberg trace formula 58J50: Spectral problems; spectral geometry; scattering theory

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Type
Research Article
Information
Mathematika , Volume 59 , Issue 2 , July 2013 , pp. 269 - 301
Copyright
Copyright © 2013 University College London 

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