Abstract
In this paper, the eigenvalue problem of the Cauchy-Riemann operator with nonlocalboundary conditions is reduced to a singular integral equation. Regularization of the singularintegral equation was carried out according to the scheme of S.G. Mikhlin, which index is equal tozero and the Noetherian condition is established for \(\lambda \neq 2\) . The resulting singular integral equation is reduced to the Fredholm linearintegral 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 regularboundary 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-localin nature, and similar problems for the Cauchy-Riemann operator were described by M. Otelbaevand A.N. Shynybekov in 1982, which gives a description of general regular boundary valueproblems for the Cauchy-Riemann operator having the property that zero belongs to the resolventset of the operator, that is, zero is a regular point where it has a non-empty resolvent set. Whenreducing the original spectral problem for the Cauchy-Riemann operator with regular (withnonlocal) boundary conditions to a singular integral equation, residues were calculated at allsingular points, in particular at essential singular points, as well as first-order poles to bring theequation under study to the canonical form. The spectral problem for the Cauchy-Riemannoperator with homogeneous boundary conditions of the Dirichlet problem type is Volterra, theresult of which was published in the early joint works of the author with B.E. Kanguzhin.
