Topics in spectral geometry

• Verfasst von: Levitin, Michael [VerfasserIn]
• Verfasst von: Mangoubi, Dan [VerfasserIn]
• Verfasst von: Polterovich, Iosif [VerfasserIn]
• Titel: Topics in spectral geometry
• Verf.angabe: Michael Levitin, Dan Mangoubi, Iosif Polterovich
• Verlagsort: Providence, Rhode Island
• Verlag: American Mathematical Society
• E-Jahr: 2023
• Jahr: [2023]
• Umfang: 1 Online-Ressource (xviii, 325 Seiten)
• Illustrationen: Illustrationen, Diagramme
• Gesamttitel/Reihe: Graduate studies in mathematics ; 237
• Fussnoten: Description based on publisher supplied metadata and other sources
• ISBN: 978-1-4704-7549-9
• Schlagwörter: (s)Spektralgeometrie / (s)Laplace-Operator / (s)Eigenwert / (s)Variationsprinzip / (s)Ungleichung / (s)Nodal Set / (s)Randwertproblem
• Datenträger: Online-Ressource
• Sprache: eng
• Bibliogr. Hinweis: Erscheint auch als : Druck-Ausgabe: Levitin, Michael, 1963 - : Topics in spectral geometry. - Providence, Rhode Island : American Mathematical Society, 2023. - xviii, 325 Seiten
• RVK-Notation: SK 350
• Sach-SW: Operator theory -- General theory of linear operators -- Eigenvalue problems
• Sach-SW: Global analysis, analysis on manifolds -- Calculus on manifolds; nonlinear operators -- Spectral theory; eigenvalue problems
• Sach-SW: Global analysis, analysis on manifolds -- Partial differential equations on manifolds; differential operators -- Spectral problems; spectral geometry; scattering theory
• Sach-SW: Global analysis, analysis on manifolds -- Partial differential equations on manifolds; differential operators -- Isospectrality
• Sach-SW: Numerical analysis -- Partial differential equations, boundary value problems -- Eigenvalue problems
• K10plus-PPN: 1870716612
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

It is remarkable that various distinct physical phenomena, such as wave propagation, heat diffusion, electron movement in quantum mechanics, oscillations of fluid in a container, can be described using the same differential operator, the Laplacian. Spectral data (i.e., eigenvalues and eigenfunctions) of the Laplacian depend in a subtle way on the geometry of the underlying object, e.g., a Euclidean domain or a Riemannian manifold, on which the operator is defined. This dependence, or, rather, the interplay between the geometry and the spectrum, is the main subject of spectral geometry. Its roots can be traced to Ernst Chladni's experiments with vibrating plates, Lord Rayleigh's theory of sound, and Mark Kac's celebrated question "Can one hear the shape of a drum?" In the second half of the twentieth century spectral geometry emerged as a separate branch of geometric analysis. Nowadays it is a rapidly developing area of mathematics, with close connections to other fields, such as differential geometry, mathematical physics, partial differential equations, number theory, dynamical systems, and numerical analysis.This book can be used for a graduate or an advanced undergraduate course on spectral geometry, starting from the basics but at the same time covering some of the exciting recent developments which can be explained without too many prerequisites.

Abstract

Cover -- Title page -- Contents -- Preface -- Introduction -- Overview -- Plan of the book -- Possible courses based on this book -- What is not in this book: Some further reading -- Chapter 1. Strings, drums, and the Laplacian -- 1.1. Basic examples -- 1.2. The Laplacian on a Riemannian manifold -- Chapter 2. The spectral theorems -- 2.1. Weak spectral theorems -- 2.2. Elliptic regularity and strong spectral theorems -- Chapter 3. Variational principles and applications -- 3.1. Variational principles for Laplace eigenvalues -- 3.2. Consequences of variational principles -- 3.3. Weyl's law and Pólya's conjecture -- Chapter 4. Nodal geometry of eigenfunctions -- 4.1. Courant's nodal domain theorem -- 4.2. Density of nodal sets -- 4.3. Yau's conjecture on the volume of nodal sets -- 4.4. Nodal sets on surfaces and eigenvalue multiplicity bounds -- Chapter 5. Eigenvalue inequalities -- 5.1. The Faber-Krahn inequality -- 5.2. Cheeger's inequality and its applications -- 5.3. Upper bounds for Laplace eigenvalues -- 5.4. Universal inequalities -- Chapter 6. Heat equation, spectral invariants, and isospectrality -- 6.1. Heat equation and spectral invariants -- 6.2. Isospectral manifolds and domains -- Chapter 7. The Steklov problem and the Dirichlet-to-Neumann map -- 7.1. The Steklov eigenvalue problem -- 7.2. The Dirichlet-to-Neumann map and the boundary Laplacian -- 7.3. Steklov spectra on domains with corners -- 7.4. The Dirichlet-to-Neumann map for the Helmholtz equation -- Appendix A. A short tutorial on numerical spectral geometry -- A.1. Overview -- A.2. Learning \FF by example -- A.3. List of downloadable scripts -- Appendix B. Background definitions and notation -- B.1. Sets -- B.2. Function spaces -- B.3. Regularity of the boundary -- Image credits -- Bibliography -- Index -- Back Cover.