Spectral theory of infinite-area hyperbolic surfaces / David Borthwick.

Format:

Book

Author/Creator:

Borthwick, David, author.

Series:

Progress in mathematics (Boston, Mass.) ; v. 318.

Progress in mathematics ; volume 318

Language:

English

Subjects (All):

Riemann surfaces.

Geometry, Hyperbolic.

Spectral theory (Mathematics).

Physical Description:

xiii, 463 pages : illustrations (some color) ; 24 cm.

Edition:

Second edition.

Other Title:

Spectral theory of infinite-volume hyperbolic surfaces.

Place of Publication:

[Cham], Switzerland : Birkhäuser, Springer International Publishing, [2016].

Summary:

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.

Contents:

Introduction

Hyperbolic Surfaces

Selberg Theory for Finite-Area Hyperbolic Surfaces

Spectral Theory for the Hyperbolic Plane

Model Resolvents for Cylinders

The Resolvent

Spectral and Scattering Theory

Resonances and Scattering Poles

Growth Estimates and Resonance Bounds

Selberg Zeta Function

Wave Trace and Poisson Formula

Resonance Asymptotics

Inverse Spectral Geometry

Patterson-Sullivan Theory

Dynamical Approach to the Zeta Function

Numerical Computations

Appendix

References

Notation Guide

Index.

Notes:

First edition published with the title, Spectral theory of infinite-volume hyperbolic surfaces, in 2007.

Includes bibliographical references (pages 447-457) and index.

Other Format:

Online version:

ISBN:

9783319338750

3319338757

OCLC:

956637108

Publisher Number:

9783319338750