Vdovenko T. Singularly nonsymmetrically perturbed selfadjoint operators

The thesis is devoted to singularly perturbed selfadjoint operators with nonsymmetric rank one perturbation that is a skew projection. There is considered a new class of singularly perturbed operators. The descriptions of such operators are given in terms of resolvents. A description allowed to investigate the point spectrum of nonsymmetrically singularly rank one perturbed operators. The inverse problem is also solved, i.e., for a given set of eigenvalues and eigenvectors there is recovered the singularly nonsymmetrically rank one perturbed operator. The analog of dual pair of eigenvalues for the singularly nonsymmetrically rank one perturbed operator is found and the correspondence operator is constructed. Examples of asymmetrically perturbed Laplace operator by δ-functions and with the operator of multiplication by an independent variable are given.