Perturbation theory for the Laplacian on automorphic functions

Abstract

Let Γ ⊂ PSL(2, R ) \Gamma \subset {\text {PSL(2, }}\mathbb {R}{\text {)}} be a discrete subgroup with quotient Γ ∖ H \Gamma \backslash H of finite volume but not compact. The spectrum of the Laplacian on L 2 {L^2} automorphic functions is unstable under perturbations; however, it becomes much more manageable when the scattering frequencies are adjoined (with multiplicity equal to the order of the pole of the determinant of the scattering matrix at these points). This augmented set shows up in a natural way in a one-sided version of the Selberg trace formula and is the actual spectrum of the generator of a cut-off wave equation. Applying standard perturbation theory to this operator, it is proved that the augmented spectrum is real analytic in Teichmüller space. The same operator is used to derive Fermi’s Golden Rule in this setting. It turns out that the proper multiplicity to be attached to the Laplacian eigenvalue at 1 4 \frac {1}{4} is twice the dimension of cusp forms plus μ = tr [ Φ + I ] /2 \mu = {\text { tr}}[\Phi {\text { + }}I]{\text {/2}} ; here Φ \Phi denotes the scattering matrix at this point. It is shown that the generic value of μ \mu in the Teichmüller space of the once punctured torus and the six-times punctured sphere is zero. This is also true of the χ \chi -twisted spectral problem, where χ \chi is a character for Γ \Gamma .