MULTIPLICITY OF EIGENVALUES OF THE DIFFUSION OPERATOR WITH A SPECTRAL PARAMETER QUADRATICALLY CONTAINED IN THE BOUNDARY CONDITION

Summary: The paper considers a boundary value problem generated by a differential diffusion equation and nonseparated boundary conditions. One of the boundary conditions contains a quadratic function of the spectral parameter. The multiplicity of eigenvalues of the boundary value problem under consideration is investigated. A criterion for the multiplicity of eigenvalues and zeros of the characteristic function of a boundary value problem is obtained.The found necessary and sufficient conditions are expressed through the values of the fundamental solutions of the diffusion equation and the coefficients of the boundary conditions. Note that the results obtained can be used in the study of direct and inverse problems of spectral analysis for various differential operators. These results also play an important role in studying the structure of the spectrum, in establishing the order of arrangement of eigenvalues of boundary value problems, and in finding sufficient conditions for the reconstruction of the corresponding problems.