科研之友
微信
新浪微博
Facebook
分享链接
摘要
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) circle times 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.
- 出版日期2013-7
