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  • Yiannis N. Petridis (a1) and Morten S. Risager (a2)

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 $u_j$ 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 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.

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