Resonances of Schottky surfaces
December 12 at 16:00 – 17:00 CET
International Seminar on Automorphic Forms
Anke Pohl (University of Bremen)
The investigation of L^2-Laplace eigenvalues and eigenfunctions for hyperbolic surfaces of finite area is a classical and exciting topic at the intersection of number theory, harmonic analysis and mathematical physics. In stark contrast, for (geometrically finite) hyperbolic surfaces of infinite area, the discrete L^2-spectrum is finite. A natural replacement are the resonances of the considered hyperbolic surface, which are the poles of the meromorphically continued resolvent of the Laplacian.
These spectral entities also play an important role in number theory and various other fields, and many fascinating results about them have already been found; the generalization of Selberg’s 3/16-theorem by Bourgain, Gamburd and Sarnak is a well-known example. However, an enormous amount of the properties of such resonances, also some very elementary ones, is still undiscovered. A few years ago, by means of numerical experiments, Borthwick noticed for some classes of Schottky surfaces (hyperbolic surfaces of infinite area without cusps and conical singularities) that their sets of resonances exhibit unexcepted and nice patterns, which are not yet fully understood.
After a brief survey of some parts of this field, we will discuss an alternative numerical method, combining tools from dynamics, zeta functions, transfer operators and thermodynamic formalism, functional analysis and approximation theory. The emphasis of the presentation will be on motivation, heuristics and pictures. This is joint work with Oscar Bandtlow, Torben Schick and Alex Weisse.
The password is the first Fourier coefficient of the modular j-function (as digits).