On a Spectral Problem for the Cauchy-Riemann Operator with Regular Boundary Conditions

N. S. Imanbaev

2025

Closed access Metadata (CSL-JSON)
DOI 10.1134/s1055134425030046
Data confidence 70 medium
How this score is calculated

It is built from checkable properties of the record itself: whether the article has a DOI, how many independent sources confirmed its data, whether the required fields are filled in, and how recent the latest confirmation is. Disagreement between sources and a missing identifier lower it. A value of 80 or above means the record was published without further checking; 60 to 79 means it is published at reduced confidence, with a manual check advisable. A record verified by an editor by hand is never pushed below 90 by a recalculation. This scores our data about the article, not the article itself.

Abstract

In this paper, the eigenvalue problem of the Cauchy-Riemann operator with nonlocal boundary conditions is reduced to a singular integral equation. Regularization of the singular integral equation was carried out according to the scheme of S.G. Mikhlin, which index is equal to zero and the Noetherian condition is established for $$ \lambda \neq 2 $$ . The resulting singular integral equation is reduced to the Fredholm linear integral equation of the second kind for $$ \lambda \neq 2 $$ and $$ Re\lambda \neq 1 $$ . The condition on the spectral parameter $$ \lambda $$ is analogous to the Lopatinsky condition. A general description of regular boundary value problems for the Cauchy-Riemann differential expression was developed by J.F. Neumann, M.I. Vishik, A.A. Dezin and M. Otelbaev.The problem under consideration is non-local in nature, and similar problems for the Cauchy-Riemann operator were described by M. Otelbaev and A.N. Shynybekov in 1982, which gives a description of general regular boundary value problems for the Cauchy-Riemann operator having the property that zero belongs to the resolvent set of the operator, that is, zero is a regular point where it has a non-empty resolvent set. When reducing the original spectral problem for the Cauchy-Riemann operator with regular (with nonlocal) boundary conditions to a singular integral equation, residues were calculated at all singular points, in particular at essential singular points, as well as first-order poles to bring the equation under study to the canonical form. The spectral problem for the Cauchy-Riemann operator with homogeneous boundary conditions of the Dirichlet problem type is Volterra, the result of which was published in the early joint works of the author with B.E. Kanguzhin.