Mathematical Theory of Scattering Resonances
Seiten
2019
American Mathematical Society (Verlag)
9781470443665 (ISBN)
American Mathematical Society (Verlag)
9781470443665 (ISBN)
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Focuses on the simplest case of scattering by compactly supported potentials, and provides pointers to modern literature where more general cases are studied. The book also presents an approach to the study of resonances on asymptotically hyperbolic manifolds. The last two chapters are devoted to semiclassical methods in the study of resonances.
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.
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.
Semyon Dyatlov, University of California, Berkeley, CA, and MIT, Cambridge, MA. Maciej Zworski, University of California, Berkeley, CA.
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
Bibliography
Index.
| Erscheinungsdatum | 01.10.2019 |
|---|---|
| Reihe/Serie | Graduate Studies in Mathematics |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 1314 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Naturwissenschaften ► Physik / Astronomie | |
| ISBN-13 | 9781470443665 / 9781470443665 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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