Correct Singular Perturbations of the Laplace Operator

Abstract

The work is devoted to the study of the Laplace operator when the potential is a singular generalized function and plays the role of a singular perturbation of the Laplace operator. Abstract theorem obtained earlier by B.N.Biyarov and G.K.Abdrasheva can be applied in this case. The main purpose of the paper is studying the related spectral problems. Singular perturbations for differential operators have been studied by many authors for the mathematical substantiation of solvable models of quantum mechanics, atomic physics, and solid state physics. In all those cases, the problems were self-adjoint. In this paper, we consider non-self-adjoint singular perturbation problems. A new method has been developed that allows investigating the considered problems.