Eigenvalue problem for differential Cauchy-Riemann operator with nonlocal boundary conditions

We consider the reduced spectral problem for the Cauchy-Riemann operator with nonlocal boundary conditions to Fredholm linear integral equation of the second kind with a continuous kernel. The corresponding Fredholm determinant is defined for all spectral parameters, excepting the points: two and one. Finding zeros of the Fredholm determinant recorded in this form is inefficient, because it is not an entire function of the spectral parameter and the main part of the determinant is not separated. Moreover, we study the structure of the kernel by the method shown above, and establish that the linear Fredholm integral equation could not be solved exactly. Therefore, for its approximate solution the results of I. Akbergenov have been applied, where the estimates of the magnitude of the difference between the exact and approximate solutions of the integral equation are given, main part of the kernel is separated. In this case, the spectral parameters are described under which the nonhomogeneous boundary value problem with shift for the Cauchy-Riemann equations is solvable everywhere in the class of continuous functions on the unit circle. Moreover, the design of the approximated solution of the nonhomogeneous boundary value problem is given.

Cauchy-Riemann operator, space of continuous functions, Fredholm property, resolvent set, problem with shift, kernel, Fredholm determinant