S. A. Kuzhel

Non-self-adjoint Schrödinger operators $A_{\mathbf{T}}$ which correspond to non-symmetric zero-range potentials are investigated. For a given $A_{\mathbf{T}}$, a description of non-real eigenvalues, spectral singularities and exceptional points are obtained; the possibility of interpretation of $A_{\mathbf{T}}$ as a self-adjoint operator in a Krein space is studied, the problem of similarity of $A_{\mathbf{T}}$ to a self-adjoint operator in a Hilbert space is solved.

$J$-self-adjoint extensions of the Phillips symmetric operator $S$ are %\break studied. The concepts of stable and unstable $C$-symmetry are introduced in the extension theory framework. The main results are the following: if ${A}$ is a $J$-self-adjoint extension of $S$, then either $\sigma({A})=\mathbb{R}$ or $\sigma({A})=\mathbb{C}$; if ${A}$ has a real spectrum, then ${A}$ has a stable $C$-symmetry and ${A}$ is similar to a self-adjoint operator; there are no $J$-self-adjoint extensions of the Phillips operator with unstable $C$-symmetry.

Finite rank point perturbations of the $p$-adic fractional differentiation operator $D^{\alpha}$ are studied. The main attention is paid to the description of operator realizations (in $L_2(\mathbb{Q}_p)$) of the heuristic expression $D^{\alpha}+\sum_{i,j=1}^{n}b_{ij}\delta_{x_i}$ in a form that is maximally adapted for the preservation of physically meaningful relations to the parameters $b_{ij}$ of the singular potential.

Finite rank perturbations of a semi-bounded self-adjoint operator $A$ are studied. Different types of finite rank perturbations (regular, singular, mixed singular) are described from a unique point of view and by the same formula with the help of quasi-boundary value spaces. As an application, a Schr\"{o}dinger operator with nonlocal point interactions is considered.