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Spectral Theory of Infinite-Area Hyperbolic Surfaces

  • Provides an accessible introduction to geometric scattering theory and the theory of resonances

  • Discusses important developments such as resonance counting, analysis of the Selberg zeta function, and the Poisson formula

  • New chapters cover resolvent estimates, wave propagation, and Naud’s proof of a spectral gap for convex hyperbolic surfaces

  • Makes use of new techniques for resonance plotting that more clearly illustrate existing results of resonance distribution

Book

Part of the Progress in Mathematics book series (PM, volume 318)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. David Borthwick
    Pages 1-6
  3. David Borthwick
    Pages 7-45
  4. David Borthwick
    Pages 63-79
  5. David Borthwick
    Pages 81-98
  6. David Borthwick
    Pages 99-119
  7. David Borthwick
    Pages 121-142
  8. David Borthwick
    Pages 143-176
  9. David Borthwick
    Pages 177-212
  10. David Borthwick
    Pages 213-246
  11. David Borthwick
    Pages 247-267
  12. David Borthwick
    Pages 269-296
  13. David Borthwick
    Pages 297-318
  14. David Borthwick
    Pages 319-368
  15. David Borthwick
    Pages 369-396
  16. David Borthwick
    Pages 397-414
  17. Back Matter
    Pages 415-463

About this book

Introduction

This text introduces geometric spectral theory in the context of infinite-area Riemann surfaces, providing a comprehensive account of the most recent developments in the field. For the second edition the context has been extended to general surfaces with hyperbolic ends, which provides a natural setting for development of the spectral theory while still keeping technical difficulties to a minimum.  All of the material from the first edition is included and updated, and new sections have been added.

Topics covered include an introduction to the geometry of hyperbolic surfaces, analysis of the resolvent of the Laplacian, scattering theory, resonances and scattering poles, the Selberg zeta function, the Poisson formula, distribution of resonances, the inverse scattering problem, Patterson-Sullivan theory, and the dynamical approach to the zeta function.  The new sections cover the latest developments in the field, including the spectral gap, resonance asymptotics near the critical line, and sharp geometric constants for resonance bounds.  A new chapter introduces recently developed techniques for resonance calculation that illuminate the existing results and conjectures on resonance distribution.

The spectral theory of hyperbolic surfaces is a point of intersection for a great variety of areas, including quantum physics, discrete groups, differential geometry, number theory, complex analysis, and ergodic theory.  This book will serve as a valuable resource for graduate students and researchers from these and other related fields. 

Review of the first edition:

"The exposition is very clear and thorough, and essentially self-contained; the proofs are detailed...The book gathers together some material which is not always easily available in the literature...To conclude, the book is certainly at a level accessible to graduate students and researchers from a rather large range of fields. Clearly, the reader...would certainly benefit greatly from it." (Colin Guillarmou, Mathematical Reviews, Issue 2008 h)

Keywords

Complex Analysis Hyperbolic Surface Resonance Theory Scattering Theory Spectral Gap Spectral Theory

Authors and affiliations

  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

About the authors

David Borthwick is Professor and Director of the Graduate Studies Department of Mathematics and Computer Science at Emory University, Georgia, USA.

Bibliographic information

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Finance, Business & Banking

Reviews

"The exposition is very clear and thorough, and essentially self-contained; the proofs are detailed...The book gathers together some material which is not always easily available in the literature...To conclude, the book is certainly at a level accessible to graduate students and researchers from a rather large range of fields. Clearly, the reader...would certainly benefit greatly from it." (Colin Guillarmou, Mathematical Reviews, Issue 2008 h)