Multiparameter Eigenvalue Problems and Their Applications in Electrodynamics

Abstract

A nonlinear n-parametric eigenvalue problem called the problem P is considered. In addition to n spectral parameters, the problem P depends on n2 numerical parameters; for zero values of these parameters, the problem splits into n linear problems \( {P}_i^0,i=\overline{1,n} \). To the problem P, one can assign n nonlinear problems Pi, which, in particular, have solutions that are not related to the solutions of the problems \( {P}_i^0 \). The problems Pi are treated in this work as “nonperturbed” problems. Using the properties of eigenvalues of the problems Pi, we prove the existence of eigenvalues of the problem P; some of these eigenvalues are not related to solutions of the problems \( {P}_i^0 \).