Automorphic spectrum and Fermi's golden rule
type: Item
instance of: scholarly article
title: Automorphic spectrum and Fermi's golden rule
zbMATH Open document ID: 0809.30035
DOI: 10.1007/BF02790224
author: Ralph S. Phillips
author: Peter Sarnak
published in: Journal d'Analyse Mathématique
publication date: 1994-10-18T00:00:00Z
review text: Let \(\Gamma < \text{PSL}_ 2 (\mathbb{R})\) be a non-cocompact cofinite discrete subgroup. One of the major unsolved problems concerning the spectrum of the Laplacian on \(\Gamma \backslash \mathbb{H}\) is that of the existence of \(L^ 2\) eigenfunctions. The authors proved in an earlier paper [Invent. Math. 80, 339-364 (1985; Zbl 0558.10017)] that the eigenvalues of \(\Gamma\) are highly unstable under deformations of \(\Gamma\). This lead to what is now called the Phillips-Sarnak conjecture. In the paper under review the authors report on their recent work [J. Am. Math. Soc. 5, No. 1, 1-32 (1992; Zbl 0743.30039)]. In particular, they analyze in a more precise way the nature of the variation of the spectrum of \(\Delta\). As a result, they obtain a formula analogous to the classical Fermi Golden Rule which explains what is behind the dissolving of eigenvalues under distortion. Besides the results of the above mentioned paper the authors give some motivation based on the classical case of perturbations of the Schrödinger operator.
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 11F72
zbMATH DE Number: 640321
zbMATH Keywords: automorphic
zbMATH Keywords: spectrum
zbMATH Keywords: scattering theory
reviewed by: Jürgen Elstrodt
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02790224
OpenAlex ID: W2003742140
description: scientific article; zbMATH DE number 640321