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Eigenfunctions of the Laplacian on a Riemannian Manifold

Steve Zelditch Northwestern University, Evanston, IL
A co-publication of the AMS and CBMS
Available Formats:
Softcover ISBN: 978-1-4704-1037-7
Product Code: CBMS/125
List Price: $79.00 MAA Member Price:$71.10
AMS Member Price: $63.20 Electronic ISBN: 978-1-4704-4344-3 Product Code: CBMS/125.E List Price:$79.00
MAA Member Price: $71.10 AMS Member Price:$63.20
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $118.50 MAA Member Price:$106.65
AMS Member Price: $94.80 Click above image for expanded view Eigenfunctions of the Laplacian on a Riemannian Manifold Steve Zelditch Northwestern University, Evanston, IL A co-publication of the AMS and CBMS Available Formats:  Softcover ISBN: 978-1-4704-1037-7 Product Code: CBMS/125  List Price:$79.00 MAA Member Price: $71.10 AMS Member Price:$63.20
 Electronic ISBN: 978-1-4704-4344-3 Product Code: CBMS/125.E
 List Price: $79.00 MAA Member Price:$71.10 AMS Member Price: $63.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$118.50 MAA Member Price: $106.65 AMS Member Price:$94.80
• Book Details

CBMS Regional Conference Series in Mathematics
Volume: 1252017; 394 pp
MSC: Primary 34; 35; 53; 58;

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.

Readership

Graduate students and researchers interested in analysis related to spectral theory and eigenfunctions of Laplacians on Riemannian manifolds.

• Table of Contents

• Chapters
• 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
• Additional Material

• Reviews

• The exposition is very clear and elegant, although it reaches rather technical results.

G. V. Rozenblum, Mathematical Reviews
• We have a very serious work of scholarship, covering (if you'll pardon the pun) quite a spectrum of analysis (largely of the hard kind, as opposed to the soft kind), and useful to many audiences, from advanced students to experienced insiders. It's also distinctive as a springboard to any number of deeper studies flowing from the material in the book's individual chapters. It promises to be a very valuable resource.

Michael Berg, MAA Reviews
• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 1252017; 394 pp
MSC: Primary 34; 35; 53; 58;

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.

Readership

Graduate students and researchers interested in analysis related to spectral theory and eigenfunctions of Laplacians on Riemannian manifolds.

• Chapters
• 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
• The exposition is very clear and elegant, although it reaches rather technical results.

G. V. Rozenblum, Mathematical Reviews
• We have a very serious work of scholarship, covering (if you'll pardon the pun) quite a spectrum of analysis (largely of the hard kind, as opposed to the soft kind), and useful to many audiences, from advanced students to experienced insiders. It's also distinctive as a springboard to any number of deeper studies flowing from the material in the book's individual chapters. It promises to be a very valuable resource.

Michael Berg, MAA Reviews
Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
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