Multiparameter eigenvalue problems and their applications in electrodynamics

D. V. Valovik ; V. Yu. Kurseeva

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 3, Tome 172 (2019), pp. 9-29

Résumé

A nonlinear $n$-parametric eigenvalue problem called the problem $P$ is considered. In addition to $n$ spectral parameters, the problem $P$ depends on $n^2$ numerical parameters; for zero values of them, it splits into $n$ linear problems $P_i^0$, $i=\overline{1,n}$. To the problem $P$, one can assign $n$ other nonlinear problems $P_i$, which, in particular, have solutions that are not related to the solutions of the problems $P_i^0$. The problems $P_i$ are treated in this work as “nonperturbed” problems. Using the properties of eigenvalues of the problems $P_i$, we prove the existence of eigenvalues of the problem $P$; some of these eigenvalues are not related to solutions of the problems $P_i^0$.

Détail

Keywords: nonlinear Sturm–Liouville-type problem, multiparameter eigenvalue problem, method of integral dispersion equations.
Mots-clés : perturbation method