On spectral question of the Cauchy-Riemann operator with homogeneous boundary value conditions

Abstract

In this paper we consider the eigenvalue problem for the Cauchy - Riemann operator with homogeneous Dirichlet type boundary conditions. The statement of the problem is justified to the theorem of M. Otelbaev and A.N. Shynybekov, which implies the correctness of the considered problem. As an example, non - local boundary conditions and Bitsadze - Samarskii type boundary conditions are given. It is taken into account that the above spectral problem for a differential Cauchy - Riemann operator with homogeneous boundary conditions of the Dirichlet type type is reduced to a singular integral, also reduces to a linear integral equation of the second kind with a continuous kernel. And it is also taken into account that the index of the singular integral equation is zero and the Noetherian condition is obtain. It is proved that the considered spectral problem does not have eigenvalues, that is, for any complex ?, has only the zero solution and thus the Cauchy - Riemann spectral problem is a Volterra problem.