# S. A. Kuzhel

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### Schrödinger operators with non-symmetric zero-range potentials

Methods Funct. Anal. Topology **20** (2014), no. 1, 34-49

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.

### On $J$-self-adjoint extensions of the Phillips symmetric operator

S. Kuzhel, O. Shapovalova, L. Vavrykovych

Methods Funct. Anal. Topology **16** (2010), no. 4, 333-348

$J$-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.

### $p$-Adic fractional differentiation operator with point interactions

Methods Funct. Anal. Topology **13** (2007), no. 2, 169-180

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 self-adjoint perturbations

Methods Funct. Anal. Topology **12** (2006), no. 3, 243-253

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.

### Nonlocal perturbations of the radial wave equation. Lax-Phillips approach

Methods Funct. Anal. Topology **8** (2002), no. 2, 59-68

### On elements of scattering theory for abstract Shrödinger equation. Lax-Phillips approach

Methods Funct. Anal. Topology **7** (2001), no. 2, 13-21

### About dependence of the Lax-Phillips scattering matrix on choice of incoming and outgoing subspaces

Methods Funct. Anal. Topology **7** (2001), no. 1, 45-52

### On inverse problem in the Lax-Phillips scattering theory for a class of second-order operator-differential equations

Methods Funct. Anal. Topology **5** (1999), no. 2, 40-43

### On some properties of abstract wave equation

Methods Funct. Anal. Topology **3** (1997), no. 1, 82-88

### On some properties of unperturbed operators

Methods Funct. Anal. Topology **2** (1996), no. 3, 78-84