On calculation of eigenvalues and eigenfunctions of a discrete operator with a nuclear resolvent perturbed by a bounded operator

A. I. Sedov

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 11 (2019) no. 1, pp. 16-23

Résumé

The problem of calculating eigenvalues and eigenfunctions of a perturbed linear self-adjoint operator with a nuclear resolvent, perturbed by a bounded operator operating in a separable Hilbert space, is being considered. In order to solve the problem, the method of regularized traces proposed by V.A. Sadovnichy and V.V. Dubrovsky and developed by their followers is used. The classical method of regularized traces for enhancement of calculations’ accuracy assumes calculation of several terms of a series. The complexity of calculation of each subsequent term of a series non-linearly increases. Alteration of the classical method, which is proposed in this work, leads to another series, the rate of convergence of which is significantly higher, which allows decreasing the number of terms of the series that are used for calculation. Developing the proposed method, there are formulas for calculation of Fourier coefficients for expansion of perturbed eigenfunctions in a series by non-perturbed ones provided in the article. Inverse Vandermonde matrix is used for calculation of first eigenfunctions. Assessments of series remainders are given.

Détail

Keywords: eigenvalues, eigenfunctions, kernel operator, perturbed operator.

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