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Progress in Mathematics

Spectral Theory of Infinite-Area Hyperbolic Surfaces

Authors: Borthwick, David

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  • 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
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Hardcover $139.99
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Softcover $139.99
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  • ISBN 978-3-319-81622-7
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About this book

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)


About the authors

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

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)

Table of contents (16 chapters)

Table of contents (16 chapters)

Buy this book

eBook $109.00
price for USA in USD
  • ISBN 978-3-319-33877-4
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $139.99
price for USA in USD
  • ISBN 978-3-319-33875-0
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually dispatched within 3 to 5 business days.
Softcover $139.99
price for USA in USD
  • ISBN 978-3-319-81622-7
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Spectral Theory of Infinite-Area Hyperbolic Surfaces
Authors
Series Title
Progress in Mathematics
Series Volume
318
Copyright
2016
Publisher
Birkhäuser Basel
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-33877-4
DOI
10.1007/978-3-319-33877-4
Hardcover ISBN
978-3-319-33875-0
Softcover ISBN
978-3-319-81622-7
Series ISSN
0743-1643
Edition Number
2
Number of Pages
XIII, 463
Number of Illustrations
27 b/w illustrations, 37 illustrations in colour
Topics

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