Non self-adjoint well-defined restrictions and extensions with real spectrum

In this paper we study the spectral properties of relatively bounded well-defined perturbations of the well-defined restrictions and extensions. The work is devoted to the study of the similarity of a well-defined restriction to some self-adjoint operator in the case when the minimal operator is symmetric. We show that the system of eigenvectors forms a Riesz basis in the case of discrete spectrum. The resulting theorem is applied to the Sturm-Liouville operator and the Laplace operator.
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. For the Sturm-Liouville operator with a potential from the Sobolev space $W^{\alpha}_2 [0, 1]$ with $-1 \leq\alpha\leq0$, the Riesz basis property of the system of eigenvectors in the Hilbert space $L_2 (0, 1)$ was proved. In all those cases, the problems were self-adjoint. In this paper, we consider non-self-adjoint singular perturbation problems for the Sturm-Liouville operator with a potential from the Sobolev space $W^{\alpha}_2 [0, 1]$ with $-2 \leq\alpha\leq0$. We also obtained a similar result for the Laplace operator. A new method has been developed that allows investigating the considered problems. 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 the considered Hilbert space.