Completeness of the Weak Eigenfunctions of one Boundary-Value-Transmission Problem

Recently, the basis properties and eigenfunction expansions in various function spaces of the eigenfunction of the regular boundary value problems with spectral parameter in the boundary conditions have been investigated by many mathematicans. However in different areas of applied mathematics and physics many problems arise in the form of singular boundary value problems involving transmission conditions at the interior singular points. Such problems are called boundary-value-transmission problems. For example, some boundary value problems with transmission conditions arise in heat and mass transfer problems, in vibrating string problems when the string loaded with additional point masses, in diffraction problems. It is not clear how the extend the classical methods to a problem with additional transmission conditions. Another major diffuculty lies in the completeness of the eigenfunctions, since the eigenvalues of a Sturm-Liouville problems with transmission conditions may not have an asymptotic expansion. This study devoted to the investigation of the Sturm-Liouville type boundary-value problems with supplementary transmission conditions. We introduce a new concept, so-called generalized eigenfunctions and study the completeness and basis properties of systems of generalized eigenfunctions for the problem under consideration.