Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
Mathematical Theory of Scattering Resonances
 
Semyon Dyatlov University of California, Berkeley, CA and MIT, Cambridge, MA
Maciej Zworski University of California, Berkeley, CA
Mathematical Theory of Scattering Resonances
Mathematical Theory of Scattering Resonances
Hardcover ISBN:  978-1-4704-4366-5
Product Code:  GSM/200
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN:  978-1-4704-5313-8
Product Code:  GSM/200.E
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $60.00
Hardcover ISBN:  978-1-4704-4366-5
eBook: ISBN:  978-1-4704-5313-8
Product Code:  GSM/200.B
List Price: $210.00$172.50
MAA Member Price: $189.00$155.25
AMS Member Price: $168.00$138.00
Mathematical Theory of Scattering Resonances
Click above image for expanded view
Mathematical Theory of Scattering Resonances
Mathematical Theory of Scattering Resonances
Semyon Dyatlov University of California, Berkeley, CA and MIT, Cambridge, MA
Maciej Zworski University of California, Berkeley, CA
Hardcover ISBN:  978-1-4704-4366-5
Product Code:  GSM/200
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN:  978-1-4704-5313-8
Product Code:  GSM/200.E
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $60.00
Hardcover ISBN:  978-1-4704-4366-5
eBook ISBN:  978-1-4704-5313-8
Product Code:  GSM/200.B
List Price: $210.00$172.50
MAA Member Price: $189.00$155.25
AMS Member Price: $168.00$138.00
  • Book Details
     
     
    Graduate Studies in Mathematics
    Volume: 2002019; 634 pp
    MSC: Primary 58; 35; 34; 81;

    Scattering resonances generalize bound states/eigenvalues for systems in which energy can scatter to infinity. A typical resonance has a rate of oscillation (just as a bound state does) and a rate of decay. Although the notion is intrinsically dynamical, an elegant mathematical formulation comes from considering meromorphic continuations of Green's functions. The poles of these meromorphic continuations capture physical information by identifying the rate of oscillation with the real part of a pole and the rate of decay with its imaginary part. An example from mathematics is given by the zeros of the Riemann zeta function: they are, essentially, the resonances of the Laplacian on the modular surface. The Riemann hypothesis then states that the decay rates for the modular surface are all either \(0\) or \(\frac14\). An example from physics is given by quasi-normal modes of black holes which appear in long-time asymptotics of gravitational waves.

    This book concentrates mostly on the simplest case of scattering by compactly supported potentials but provides pointers to modern literature where more general cases are studied. It also presents a recent approach to the study of resonances on asymptotically hyperbolic manifolds. The last two chapters are devoted to semiclassical methods in the study of resonances.

    Readership

    Graduate students and researchers interested in scattering resonances.

  • Table of Contents
     
     
    • Chapters
    • Introduction
    • Potential scattering
    • Scattering resonances in dimension one
    • Scattering resonances in odd dimensions
    • Geometric scattering
    • Black box scattering in $\mathbb {R}^n$
    • Scattering on hyperbolic manifolds
    • Resonances in the semiclassical limit
    • Resonance-free regions
    • Resonances and trapping
    • Appendices
    • Notation
    • Spectral theory
    • Fredholm theory
    • Complex analysis
    • Semiclassical analysis
  • Reviews
     
     
    • This is an up to date account of modern mathematical scattering theory with an emphasis on the deep interplay between the location of the scattering poles or resonances, and the underlying dynamics and geometry. The masterful exposition reflects the authors' significant roles in shaping this very active field. A must read for researchers and students working in scattering theory or related areas.

      Peter Sarnak, Institute for Advanced Study
    • This is a very broad treatise of the modern theory of scattering resonances, beautifully written with a wealth of important mathematical results as well as applications, motivations and numerical and experimental illustrations. For experts, it will be a basic reference and for non-experts and graduate students an appealing and quite accessible introduction to a fascinating field with multiple connections to other branches of mathematics and to physics.

      Johannes Sjöstrand, Université de Bourgogne
    • Resonance is the Queen of the realm of waves. No other book addresses this realm so completely and compellingly, oscillating effortlessly between illustration, example, and rigorous mathematical discourse. Mathematicians will find a wonderful array of physical phenomena given a solid intuitive and mathematical foundation, linked to deep theorems. Physicists and engineers will be inspired to consider new realms and phenomena. Chapters travel between motivation, light mathematics, and deeper mathematics, passing the baton from one to the other and back in a way that these authors are uniquely qualified to do.

      Eric J. Heller, Harvard University
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2002019; 634 pp
MSC: Primary 58; 35; 34; 81;

Scattering resonances generalize bound states/eigenvalues for systems in which energy can scatter to infinity. A typical resonance has a rate of oscillation (just as a bound state does) and a rate of decay. Although the notion is intrinsically dynamical, an elegant mathematical formulation comes from considering meromorphic continuations of Green's functions. The poles of these meromorphic continuations capture physical information by identifying the rate of oscillation with the real part of a pole and the rate of decay with its imaginary part. An example from mathematics is given by the zeros of the Riemann zeta function: they are, essentially, the resonances of the Laplacian on the modular surface. The Riemann hypothesis then states that the decay rates for the modular surface are all either \(0\) or \(\frac14\). An example from physics is given by quasi-normal modes of black holes which appear in long-time asymptotics of gravitational waves.

This book concentrates mostly on the simplest case of scattering by compactly supported potentials but provides pointers to modern literature where more general cases are studied. It also presents a recent approach to the study of resonances on asymptotically hyperbolic manifolds. The last two chapters are devoted to semiclassical methods in the study of resonances.

Readership

Graduate students and researchers interested in scattering resonances.

  • Chapters
  • Introduction
  • Potential scattering
  • Scattering resonances in dimension one
  • Scattering resonances in odd dimensions
  • Geometric scattering
  • Black box scattering in $\mathbb {R}^n$
  • Scattering on hyperbolic manifolds
  • Resonances in the semiclassical limit
  • Resonance-free regions
  • Resonances and trapping
  • Appendices
  • Notation
  • Spectral theory
  • Fredholm theory
  • Complex analysis
  • Semiclassical analysis
  • This is an up to date account of modern mathematical scattering theory with an emphasis on the deep interplay between the location of the scattering poles or resonances, and the underlying dynamics and geometry. The masterful exposition reflects the authors' significant roles in shaping this very active field. A must read for researchers and students working in scattering theory or related areas.

    Peter Sarnak, Institute for Advanced Study
  • This is a very broad treatise of the modern theory of scattering resonances, beautifully written with a wealth of important mathematical results as well as applications, motivations and numerical and experimental illustrations. For experts, it will be a basic reference and for non-experts and graduate students an appealing and quite accessible introduction to a fascinating field with multiple connections to other branches of mathematics and to physics.

    Johannes Sjöstrand, Université de Bourgogne
  • Resonance is the Queen of the realm of waves. No other book addresses this realm so completely and compellingly, oscillating effortlessly between illustration, example, and rigorous mathematical discourse. Mathematicians will find a wonderful array of physical phenomena given a solid intuitive and mathematical foundation, linked to deep theorems. Physicists and engineers will be inspired to consider new realms and phenomena. Chapters travel between motivation, light mathematics, and deeper mathematics, passing the baton from one to the other and back in a way that these authors are uniquely qualified to do.

    Eric J. Heller, Harvard University
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.