SpringerBriefs in Mathematical Physics

Inverse Spectral and Scattering Theory

An Introduction

Authors: Isozaki, Hiroshi

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  • Is the first book on multi-dimensional spectral theory and inverse scattering on non-compact manifolds
  • Serves as a comprehensive introductory guide to PDEs, with minimal preparatory knowledge of mathematics
  • Presents an abundance of topics, from boundary value problems in a bounded domain to lattice problems 
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About this book

The aim of this book is to provide basic knowledge of the inverse problems arising in various areas in mathematics, physics, engineering, and medical science. These practical problems boil down to the mathematical question in which one tries to recover the operator (coefficients) or the domain (manifolds) from spectral data. The characteristic properties of the operators in question are often reduced to those of Schrödinger operators. We start from the 1-dimensional theory to observe the main features of inverse spectral problems and then proceed to multi-dimensions. The first milestone is the Borg–Levinson theorem in the inverse Dirichlet problem in a bounded domain elucidating basic motivation of the inverse problem as well as the difference between 1-dimension and multi-dimension. The main theme is the inverse scattering, in which the spectral data is Heisenberg’s S-matrix defined through the observation of the asymptotic behavior at infinity of solutions. Significant progress has been made in the past 30 years by using the Faddeev–Green function or the complex geometrical optics solution by Sylvester and Uhlmann, which made it possible to reconstruct the potential from the S-matrix of one fixed energy. One can also prove the equivalence of the knowledge of S-matrix and that of the Dirichlet-to-Neumann map for boundary value problems in bounded domains. We apply this idea also to the Dirac equation, the Maxwell equation, and discrete Schrödinger operators on perturbed lattices. Our final topic is the boundary control method introduced by Belishev and Kurylev, which is for the moment the only systematic method for the reconstruction of the Riemannian metric from the boundary observation, which we apply to the inverse scattering on non-compact manifolds. We stress that this book focuses on the lucid exposition of these problems and mathematical backgrounds by explaining the basic knowledge of functional analysis and spectral theory, omitting the technical details in order to make the book accessible to graduate students as an introduction to partial differential equations (PDEs) and functional analysis. 

Table of contents (5 chapters)

Table of contents (5 chapters)

Buy this book

eBook $44.99
price for Mexico
  • ISBN 978-981-15-8199-1
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase Institutional customers should get in touch with their account manager
Softcover $59.99
price for Mexico
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Bibliographic Information

Bibliographic Information
Book Title
Inverse Spectral and Scattering Theory
Book Subtitle
An Introduction
Authors
Series Title
SpringerBriefs in Mathematical Physics
Series Volume
38
Copyright
2020
Publisher
Springer Singapore
Copyright Holder
The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd.
eBook ISBN
978-981-15-8199-1
DOI
10.1007/978-981-15-8199-1
Softcover ISBN
978-981-15-8198-4
Series ISSN
2197-1757
Edition Number
1
Number of Pages
XII, 130
Number of Illustrations
2 b/w illustrations
Topics