Fraczek M.S. Selberg Zeta Functions and Transfer Operators: An Experimental Approach to Singular Perturbations

Springer International Publishing AG, 2017. — 363 p. — (Lecture Notes in Mathematics 2139) — ISBN: 3319512943.In recent years the application of the transfer operator method in the study of Selberg zeta functions and the spectral theory of hyperbolic spaces has made significant progress, in both analytical investigations and numerical investigations. We consider transfer operators for the geodesic flow on surfaces of constant negative curvature, therefore systems where a particle is moving freely on such a surface with constant velocity. We introduce a method for approximating transfer operators by finite dimensionalmatrices. These approximations of transfer operators allow us to compute numerically both eigenvalues and eigenfunctions of these transfer operators. Selberg zeta functions can be expressed in terms of Fredholm determinants of transfer operators. The zeros of these functions are related to both the discrete spectrum and resonances of the hyperbolic Laplace-Beltrami operator, where the resonances are the poles of the scattering determinant. We will study both singular and non-singular perturbations of the Laplace-Beltrami operator numerically by studying the zeros of the Selberg zeta function under such perturbations. Further, there is a surprising connection between the eigenfunctions of the hyperbolic Laplac-Beltrami operator, the so-called Maass wave forms, and certain eigenfunctions of the transfer operator. These eigenfunctions of the transfer operator are solutions of the Lewis three-term functional equation, which on the other hand are related toMaass wave forms through a certain integral transform.We will also discuss certain symmetries of a transfer operator and how these symmetries are related to involutions of the Maass wave forms.Preliminaries
The Gamma Function and the Incomplete Gamma Functions
The Hurwitz Zeta Function and the Lerch Zeta Function
Computation of the Spectra and Eigenvectors of Large Complex Matrices
The Hyperbolic Laplace-Beltrami Operator
Transfer Operators for the Geodesic Flow on Hyperbolic Surfaces
Numerical Results for Spectra and Traces of the Transfer Operator for Character Deformations
Investigations of Selberg Zeta Functions Under Character Deformations
Concluding Remarks
Appendixes
Computational Aspects of the Transfer Operator for the Kac-Baker Model
Project MORPHEUS
The Representatives of
The Transfer Operator for
The Zeros and Poles of the Selberg Zeta Function for Arithmetic and Arithmetic