Hangzhou Lectures on Eigenfunctions of the Laplacian (eBook)

Christopher D. Sogge (Autor)

eBook Download: PDF | EPUB

2014
208 Seiten
Princeton University Press (Verlag)
978-1-4008-5054-9 (ISBN)

Infos

Christopher D. Sogge is the J. J. Sylvester Professor of Mathematics at Johns Hopkins University. He is the author of Fourier Integrals in Classical Analysis and Lectures on Nonlinear Wave Equations.

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Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. Christopher Sogge gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace-Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. Sogge shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic.Sogge begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. He avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. The author also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, Sogge demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.

Produkt-Details

• Erscheint lt. Verlag: 10.3.2014
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Zusatzinfo: 1 line illus.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: absolute value • Addition • Approximation • arc length • Asymptote • bijection • Calculation • Canonical transformation • Cauchy problem • Cauchy–Schwarz inequality • Cauchy's Integral Formula • chain rule • Change of variables • Characteristic function (probability theory) • Cholesky decomposition • classical ergodicity • compact manifolds • Compact space • Constant function • continuous function • convolution • coordinate system • corollary • cotangent bundle • Counting • Curvature • d'Alembertian • Derivative • diffeomorphism • Differential operator • Dimension • Dimension (vector space) • Division by zero • dual space • Duistermaat-Guillemin theorem • Egorov's theorem • Eigenfunction • eigenfunctions • eigenvalues • Eigenvalues and Eigenvectors • elliptic regularity estimates • Equation • Ergodicity • ergodic theory • Error Term • estimation • Euclidean Laplacian • Euclidean space • Euler–Lagrange equation • Euler's formula • exponential function • Fourier inversion theorem • Fourier transform • Friedrichs quantization • Fundamental domain • fundamental solution • gamma function • Gauss's lemma (number theory) • geodesic • Geodesic flow • geodesics • Hadamard parametrix • hamiltonian mechanics • Heaviside step function • high frequency eigenfunctions • Integration by parts • invariant measure • Jacobian matrix and determinant • Kronecker delta • LaplaceЂeltrami operators • Laplacian • Lebesgue measure • limit theorems • Linear combination • Linear map • Local coordinates • Manifolds • Mathematical Induction • microlocal analysis • Minkowski space • Natural number • Nonpositive curvature • normal coordinates • Open set • Orthonormal basis • Oscillatory integral • Parametrix • partition of unity • Periodic function • periodic geodesics • Pointwise • Polar coordinate system • probability measure • Projection (linear algebra) • pseudo-differential operator • pseudodifferential operators • pullback • Pullback (category theory) • Pullback (differential geometry) • quadratic form • Quantity • Quantum Chaos • quantum ergodicity • Quantum limit • quantum state • Riemannian Geometry • Riemannian manifold • riemannian manifolds • sectional curvature • Self-adjoint • sharp Weyl formula • shrinking spectral bands • Singularities • Smoothness • Sobolev Space • Special case • Spectral Asymptotics • Spherical Harmonics • square root • stationary phase • Stationary point • subsequence • Subset • sup-norm estimates • Support (mathematics) • Tangent Space • Tangent vector • Theorem • Topology • torus • trace estimates • Translational symmetry • Variable (mathematics) • Vector field • Volume element • Wave Equations • wave front sets • Weyl formula • Weyl law
• ISBN-10: 1-4008-5054-1 / 1400850541
• ISBN-13: 978-1-4008-5054-9 / 9781400850549