Energy Methods in Self-Adjoint Eigenvalue Problems. I. Variational Theory of the Spectrum

摘要

An essentially self-contained elementary account, from a unified variational point of view, is given of the theory of self-adjoint eigenvalue problems with discrete spectra, governed by linear differential equations of the form M(y) =lambda N(y). The theory is directly relevant for the various types of approximate energy methods applied in such problems. Included herein are statements and proofs of the variational, minimum, and maximum-minimum characterization of the eigenvalues in all modes. Theorems based on both the Rayleigh quotient and the energy quotient, including the role of natural boundary conditions, are developed. In addition, existence proofs, and discussion and proofs of completeness in both the N-norm and M-norm are given. (Author)