Asymptotics of the spectra of the Dirichlet and Dirichlet-Neumann problems for the Sturm-Liouville equation with integral perturbation

TL;DR

This paper derives precise asymptotic formulas for eigenvalues of Sturm-Liouville problems with integral perturbations, revealing how potential and kernel Fourier coefficients influence spectral behavior.

Contribution

It provides new sharp asymptotic formulas for eigenvalues of Sturm-Liouville problems with convolution kernel perturbations, including estimates for remainder terms.

Findings

Derived asymptotic formulas involving Fourier coefficients.

Established estimates for remainder terms considering eigenvalue growth.

Extended results to the case with zero convolution kernel.

Abstract

The article studies the Dirichlet and Dirichlet-Neumann problems for the Sturm-Liouville equation perturbed by an integral operator with a convolution kernel. Sharp asymptotic formulas for the eigenvalues of these problems are found. The formulas contain information about the Fourier coefficients of the potential and the kernel, and estimates are obtained for the remainder terms of the asymptotics, which take into account both the rate of decrease as the eigenvvalues tend to infinity and the rate of decrease as the norms of the potential and kernel tend to zero. The formulas are also new in the case of the Sturm-Liouville operator, when the convolution kernel is zero.