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BL
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| Record Number
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850253
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| Main Entry
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Zelditch, Steven,1953-
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| Title & Author
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Eigenfunctions of the Laplacian on a Riemannian manifold /\ Steve Zelditch.
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| Publication Statement
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Providence, Rhode Island :: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, with the support from the National Science Foundation,, [2017]
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| Series Statement
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Regional conference series in mathematics ;; number 125
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| Page. NO
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xiv, 394 pages :: illustrations ;; 26 cm
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| ISBN
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1470410370
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: 9781470410377
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| Notes
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Based on the author's notes from his presentation at the NSF-CBMS Regional Conference in the Mathematical Sciences on Global Harmonic Analysis, held at University of Kentucky, June 20-24, 2011.
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Introduction -- Geometric preliminaries -- Main results -- Model spaces of constant curvature -- Local structure of eigenfunctions -- Hadamard parametrices on Riemannian manifolds -- Lagrangian distributions and Fourier integral operators -- Small time wave group and Weyl asymptotics -- Matrix elements -- Lp norms -- Quantum integrable systems -- Restriction theorems -- Nodal sets: real domain -- Eigenfunctions in the complex domain.
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| Abstract
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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 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"--Back cover.
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| Subject
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Eigenfunctions.
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Laplacian operator.
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Riemannian manifolds.
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Differential geometry-- Symplectic geometry, contact geometry-- Geodesic flows.
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| Subject
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Eigenfunctions.
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| Subject
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Global analysis, analysis on manifolds-- Partial differential equations on manifolds; differential operators-- Pseudodifferential and Fourier integral operators on manifolds.
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| Subject
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Global analysis, analysis on manifolds-- Partial differential equations on manifolds; differential operators-- Spectral problems; spectral geometry; scattering theory.
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| Subject
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Laplacian operator.
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Ordinary differential equations-- Ordinary differential operators-- Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions.
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Partial differential equations-- Elliptic equations and systems-- Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation.
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Partial differential equations-- Hyperbolic equations and systems-- Wave equation.
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Partial differential equations-- Spectral theory and eigenvalue problems-- Asymptotic distribution of eigenvalues and eigenfunctions.
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| Subject
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Riemannian manifolds.
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| Dewey Classification
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516.3/62
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| LC Classification
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QA649.Z45 2017
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| NLM classification
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34L2035P2035J0535L0553D2558J4058J50msc
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58J50.msc
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| Added Entry
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American Mathematical Society,issuing body.
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National Science Foundation (U.S.),sponsoring body.
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