On Selberg's small eigenvalue conjecture and residual eigenvalues

TL;DR

This paper establishes an equivalence between Selberg's small eigenvalue conjecture and a conjecture about residual eigenvalues, using methods from asymptotic perturbation theory and number theory.

Contribution

It demonstrates that Selberg's conjecture is equivalent to the non-existence of residual eigenvalues in a perturbed automorphic Laplacian system.

Findings

Selberg's eigenvalue conjecture is equivalent to a residual eigenvalue non-existence conjecture.

The proof combines asymptotic perturbation theory with number theory techniques.

The approach provides a new perspective on the spectral properties of automorphic Laplacians.

Abstract

We show that Selberg's eigenvalue conjecture concerning small eigenvalues of the automorphic Laplacian for congruence groups is equivalent to a conjecture about the non-existence of residual eigenvalues for a perturbed system. We prove this using a combination of methods from asymptotic perturbation theory and number theory.