Dissolving cusp forms: Higher order Fermi's Golden Rules

TL;DR

This paper develops higher order Fermi's Golden Rules to determine when embedded eigenvalues of the Laplace operator on hyperbolic surfaces dissolve into resonances, linking the process to special values of L-series and providing necessary and sufficient conditions.

Contribution

It extends the classical Fermi's Golden Rule by deriving formulas for higher order approximations and establishing precise conditions for eigenvalue dissolution into resonances.

Findings

Derived formulas for higher order approximations of eigenvalue dissolution.

Established necessary and sufficient conditions involving L-series values.

Demonstrated the conditions' implications in a specific example.

Abstract

For a hyperbolic surface embedded eigenvalues of the Laplace operator are unstable and tend to 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 2 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.