© 2017

Selberg Zeta Functions and Transfer Operators

An Experimental Approach to Singular Perturbations


  • The only book on the market which describes the evaluation of Selberg zeta functions for character deformations via the transfer operator method

  • Gives a detailed description of numerical methods and analytic theories in one book

  • Provides animations and over 50 color illustrations, helping the reader to get a better understanding

  • Gives numerical and analytical results on new phenomena related to singular perturbations of hyperbolic Laplacians


Part of the Lecture Notes in Mathematics book series (LNM, volume 2139)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Markus Szymon Fraczek
    Pages 1-23
  3. Markus Szymon Fraczek
    Pages 25-37
  4. Markus Szymon Fraczek
    Pages 39-42
  5. Markus Szymon Fraczek
    Pages 43-68
  6. Markus Szymon Fraczek
    Pages 87-127
  7. Markus Szymon Fraczek
    Pages 297-300
  8. Back Matter
    Pages 301-354

About this book


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.


Character deformation Eigenvalues and resonances of hyperbolic Laplacian Selberg zeta function Singular perturbations of hyperbolic Laplacian Transfer operator method

Authors and affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUnited Kingdom

Bibliographic information

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“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)