Eigenfunctions of the Laplacian on a Riemannian Manifold
Seiten
2017
American Mathematical Society (Verlag)
978-1-4704-1037-7 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-1037-7 (ISBN)
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Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The emphasis is on the global methods and the use of Fourier integral operator methods to analyse norms and nodal sets of eigenfunctions.
Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow.
The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain.
The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research.
Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow.
The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain.
The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research.
Steve Zelditch, Northwestern University, Evanston, IL.
Introduction
Geometric preliminaries
Main results
Model spaces of constant curvature
Local structure of eigenfunctions
Hadamard parametrics on Riemannian manifolds
Lagrangian distributions and Fourier integral operators
Small time wave group and Weyl asymptotics
Matrix elements
$L^p$ norms
Quantum integrable systems
Restriction theorems
Nodal sets: Real domain
Eigenfunctions in the complex domain
Index.
| Erscheinungsdatum | 07.01.2018 |
|---|---|
| Reihe/Serie | CBMS Regional Conference Series in Mathematics |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 525 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik | |
| ISBN-10 | 1-4704-1037-0 / 1470410370 |
| ISBN-13 | 978-1-4704-1037-7 / 9781470410377 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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