DISSOLVING OF CUSP FORMS: HIGHER‐ORDER FERMI'S GOLDEN RULES

Abstract

For a hyperbolic surface, embedded eigenvalues of the Laplace operator are unstable and tend to dissolve into scattering poles i.e. 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 into a resonance. In the framework of perturbations in character varieties, we relate the result to the special values of the -series . This is the Rankin–Selberg convolution of with , where is the antiderivative of a weight two 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.