Asymptotics of the Spectra of the Dirichlet and Dirichlet–Neumann Problems for the Sturm–Liouville Equation with an Integral Perturbation

V. N. Sivkin^ab, A. A. Shkalikov<sup>cb

a</sup> National Research University Higher School of Economics, Moscow
^b Moscow Center for Fundamental and Applied Mathematics
^c Lomonosov Moscow State University

Abstract: We study 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 for the remainder terms of the asymptotics we obtain estimates taking into account the decay rates both as the eigenvalues tend to infinity and as the norms of the potential and the kernel tend to zero. The formulas are new even in the case of the Sturm–Liouville operator, where the convolution kernel is zero.

Keywords: Sturm–Liouville operator, integro-differential operator, asymptotic formulas for eigenvalues, Hardy operator.

UDC: 517

Received: 31.05.2025

DOI: 10.4213/mzm14738

 English version: Mathematical Notes, 2025, 118:2, 356–374

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