Eigenvalue problems, spectral parameter power series, and modern applications

Abstract

"Our review is dedicated to a wide class of spectral and transmission problems arising in di?erent branches of applied physics. One of the main di?culties in studying and solving eigenvalue problems for operators with variable coe?cients consists in obtaining a corresponding dispersion relation or characteristic equa-tion of the problem in a su?ciently explicit form. Solutions of the dispersion relation are the eigenvalues of the problem. When the dispersion relation is known the eigenvalues are found numerically even for relatively simple problems with constant coe?cients because even in those cases as a rule the dispersion relation represents a transcendental equation the exact solutions of which are unknown."