Automorphic spectrum and Fermi's golden rule
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.
scientific article; zbMATH DE number 640321
scientific article