Selberg Zeta Functions and Transfer Operators: An Experimental Approach to Singular Perturbations

ISBN-13: 9783319512945 / Angielski / Miękka / 2017 / 354 str.

Markus Szymon Fraczek

Opis

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.

Recenzje

"This volume is an interesting contribution to a field that still holds a lots of secrets. It will be of great interest to experts." (Ch. Baxa, Monatshefte für Mathematik, Vol. 198 (2), June, 2022)

"What makes this book a unique is that it systematically covers effective computation of the spectral terms of the selberg trace formula, namely the eigenvectors, eigenfunctions and resonances. ... The computation of this book gives us a hint as to what actually occurs with the extremely complicated limit ... The book is self-contained, covering both the theoretical background and the numerical aspects." (Joshua S. Friedman, Mathematical Reviews, May, 2018)

Spis treści

Introduction.-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.-Appendices.-References.-Index of Notations.