The study of eigenvalues and eigenfunctions of selfadjoint elliptic differential operators such as the Laplacian and Schrodinger operators on domains and manifolds. Covers the variational (Rayleigh quotient and minimax) characterization of eigenvalues, existence of a discrete spectrum via compactness of the resolvent, orthonormal eigenfunction bases of L2, dependence of the spectrum on Dirichlet, Neumann and Robin boundary conditions, domain and coefficient monotonicity, Weyl asymptotics and Polya-type bounds, and the way spectral data governs stability and long-time behaviour of the associated heat, wave and Schrodinger evolution equations.
Lecture notes for a half-semester graduate course on spectral theory for selfadjoint partial differential operators, taught from computable examples before general theory. The spectrum of an operator…