Perturbation of a linear operator by a bilinear operator

Abstract

The problem of perturbation of the spectrum of a linear operator in the linear case has been solved long ago and successfully. Important concepts such as holomorphic families of the type $(A)$ and in the sense of Kato have also been introduced. The problems associated with perturbation of the spectrum are also well-known - problems of determining energy levels in quantum mechanics. At the same time, not always built Rayleigh--Schr\"{o}dinger series converge in the usual, and not in the asymptotic sense. In this article, the conditions for ordinary convergence of series in powers of a small parameter were found, representing solutions of one type of nonlinear problems to eigenvalues.