Selberg Zeta Functions and Transfer Operators An Experimental Approach to Singular Perturbations 1st Edition Markus Szymon Fraczek 331951296X 9783319512969

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Product details:

• ISBN 10: 331951296X
• ISBN 13: 9783319512969
• Author: Markus Szymon Fraczek

This book presents a method for evaluating Selberg zeta functions via transfer operators for the full modular group and its congruence subgroups with characters. Studying zeros of Selberg zeta functions for character deformations allows us to access the discrete spectra and resonances of hyperbolic Laplacians under both singular and non-singular perturbations. Areas in which the theory has not yet been sufficiently developed, such as the spectral theory of transfer operators or the singular perturbation theory of hyperbolic Laplacians, will profit from the numerical experiments discussed in this book. Detailed descriptions of numerical approaches to the spectra and eigenfunctions of transfer operators and to computations of Selberg zeta functions will be of value to researchers active in analysis, while those researchers focusing more on numerical aspects will benefit from discussions of the analytic theory, in particular those concerning the transfer operator method and the spectral theory of hyperbolic spaces.

Table of contents:

1. Introduction

2. Preliminaries

3. The Gamma Function and the Incomplete Gamma Functions

4. The Hurwitz Zeta Function and the Lerch Zeta Function

5. Computation of the Spectra and Eigenvectors of Large Complex Matrices

6. The Hyperbolic Laplace-Beltrami Operator

7. Transfer Operators for the Geodesic Flow on Hyperbolic Surfaces

8. Numerical Results for Spectra and Traces of the Transfer Operator for Character Deformations

9. Investigations of Selberg Zeta Functions Under Character Deformations

10. Concluding Remarks

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