Dissolving cusp forms: Higher order Fermi's Golden Rules

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.

33 pages, typos corrected, new section added

References in corpus (1)

• Automorphic forms of higher order

Cited by in corpus (2)

• Arithmetic statistics of modular symbols
• A Fermi golden rule for quantum graphs