arXiv subjects
Publications and source records attributed to B. N. Biyarov.
The description of all correct restrictions of the maximal operator are considered in a Hilbert space. A class of correct restrictions are obtained for which a similar transformation has the domain of the fixed correct restriction. The resulting theorem is applied to the study of n-order differentiation operator with singular coefficients.
The work is devoted to the study of the similarity of a correct restriction to some self-adjoint operator in the case when the minimal operator is symmetric. The resulting theorem was applied to the Sturm-Liouville operator and the Laplace operator. It is shown that the spectrum of a non self-adjoint singularly perturbed operator is real and the corresponding system of eigenvectors forms a Riesz basis.
In this paper, we establish necessary and sufficient conditions for the doubleness all of the eigenvalues, except the lowest, periodic and anti-periodic problems for Hill's equation in terms of the complex-valued potential $q(x)$.
We consider the spectral problems for the Sturm-Liouville operator generated by the Dirichlet, Neumann, Dirichlet-Neumann and Neumann-Dirichlet conditions. The necessary and sufficient condition for the coincidence of the spectrum of the Dirichlet-Neumann and Neumann-Dirichlet problems is proved. Also the necessary and sufficient condition for the coincidence of the spectrum, except zero, of the Dirichlet and Neumann problems is proved. An application to periodic and anti-periodic problems is given.
The work is devoted to the study of Laplace operator when the potential is a singular generalized function and plays the role of a singular perturbation of a Laplace operator. Abstract theorem obtained earlier by the authors B.N. Biyarov and G.K. Abdrasheva applies to this. The main purpose of the study is the spectral issue. 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 these 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.