On The Uniform Convergence Of Spectral Expansions For A Spectral Problem With A Boundary Condition Rationally Depending On The Eigenparameter

Abstract

The spectral problem −y ′′ + q(x)y = λy, 0 < x < 1, y(0) cos β = y ′ (0) sin β, 0 ≤ β < π; y ′ (1) y(1) = h(λ), is considered, where λ is a spectral parameter, q(x) is real-valued continuous function on [0, 1] and h(λ) = aλ + b − XN k=1 bk λ − ck , with the real coefficients and a ≥ 0, bk > 0, c1 < c2 < · · · < cN , N ≥ 0. The sharpened asymptotic formulae for eigenvalues and eigenfunctions of above-mentioned spectral problem are obtained and the uniform convergence of the spectral expansions of the continuous functions in terms of eigenfunctions are presented.