Spectral Theory of Partial Differential Equations - Lecture Notes

2012/03/11 by Richard S. Laugesen, Laugesen, Richard S. · 1 citation
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.AP #math.MP #math.SP

paper · pdf · doi:10.48550/arxiv.1203.2344

120 pages; 37 figures

arxiv created 2012/03/11 · arxiv updated 2012/03/13

Abstract

This mini-course of 20 lectures aims at highlights of spectral theory for self-adjoint partial differential operators, with a heavy emphasis on problems with discrete spectrum. Part I: Discrete Spectrum (ODE preview, Laplacian - computable spectra, Schroedinger - computable spectra, Discrete spectral theorem via sesquilinear forms, Laplace eigenfunctions, Natural boundary conditions, Magnetic Laplacian, Schroedinger in confining well, Variational characterizations, Monotonicity of eigenvalues, Weyl's asymptotic, Polya's conjecture, Reaction-diffusion stability, Thin fluid film stability) Part II: Continuous Spectrum (Laplacian on whole space, Schroedinger with -2sech² potential, Selfadjoint operators, Spectra: discrete and continuous, Discrete spectrum revisited)

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