# Kuzhel' S. A.

### Scattering Theory for 0-Perturbed $\mathcal{P}\mathcal{T}$ -Symmetric Operators

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1059–1079

The aim of the present work is to develop the scattering theory for 0-perturbed $\mathcal{P}\mathcal{T}$ -symmetric operators by using the Lax–Phillips method. The presence of a stable $\mathcal{C}$ -symmetry leading to the property of selfadjointness (with proper choice of the inner product) for these PTPT -symmetric operators is described in terms of the corresponding $\mathcal{C}$ -matrix (scattering matrix).

### On the theory of $\mathcal{PT}$-symmetric operatorss

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 32-49

This article develops a general theory of $\mathcal{PT}$-symmetric operators. Special attention is given to $\mathcal{PT}$-symmetric quasi-self-adjoint extensions of symmetric operator with deficiency indices 〈 2, 2 〉. For these extensions, the possibility of their interpretation as self-adjoint operators in Krein spaces is investigated, and a description of nonreal eigenvalues is given. These abstract results are applied to the Schrodinger operator with Coulomb potential on the real axis.

### Myroslav L’vovych Horbachuk (on his 70th birthday)

Adamyan V. M., Berezansky Yu. M., Khruslov E. Ya., Kochubei A. N., Kuzhel' S. A., Marchenko V. O., Mikhailets V. A., Nizhnik L. P., Ptashnik B. I., Rofe-Beketov F. S., Samoilenko A. M., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 439–442

### On infinite-rank singular perturbations of the Schrödinger operator

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 487–496

Schrodinger operators with infinite-rank singular potentials $\sum^\infty_{i,j=1}b_{i,j}(\psi_j,\cdot)\psi_i$ are studied under the condition that singular elements $\psi_j$ are $\xi_j(t)$-invariant with respect to scaling transformations in ${\mathbb R}^3$.

### Singularly perturbed self-adjoint operators in scales of Hilbert spaces

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 723–743

Finite rank perturbations of a semi-bounded self-adjoint operator $A$ are studied in the scale of Hilbert spaces associated with $A$. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and singular perturbations of $A$ by the same formula. As an application the one-dimensional Schrodinger operator with generalized zero-range potential is considered in the Sobolev space $W_2^p(R),\quad p \in N$.

### Leonid Pavlovych Nyzhnyk (on his 70-th birthday)

Berezansky Yu. M., Gorbachuk M. L., Gorbachuk V. I., Khruslov E. Ya., Kostyuchenko A. G., Kuzhel' S. A., Marchenko V. O., Samoilenko A. M., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1120-1122

### On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 679–688

For a singular perturbation $A = A_0 + \sum^n_{i, j=1}t_{ij} \langle \psi_j, \cdot \rangle \psi_i,\quad n \leq \infty$ of a positive self-adjoint operator $A_0$ with Lebesgue spectrum, the spectral analysis of the corresponding self-adjoint operator realizations $A_T$ is carried out and the scattering matrix $\mathfrak{S}_{(A_T, A_0)}(\delta)$ is calculated in terms of parameters $t_{ij}$ under some additional restrictions on singular elements $\psi_{j}$ that provides the possibility of application of the Lax -Phillips approach in the scattering theory.

### On Conditions for the Applicability of the Lax–Phillips Scattering Scheme to the Investigation of an Abstract Wave Equation

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 621-630

We find necessary and sufficient conditions under which orthogonal incoming and outgoing subspaces exist for a group of solutions of an abstract wave equation and possess an additional property of “equivalence” with respect to the operator of time reversion.

### On the inverse problem for perturbations of an abstract wave equation in the Lax-Phillips scattering scheme

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 638-647

The inverse scattering problem for perturbations of an abstract wave equation is investigated within the framework of the Lax-Phillips scattering scheme.

### On the structure of incoming and outgoing subspaces for a wave equations in $ℝ^n$

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 708-712

We investigate the structure of incoming and outgoing subspaces in the Lax-Phillips scheme for the classic wave equation in $ℝ^n$.

### On a form of the scattering matrix for ρ-perturbations of an abstract wave equations

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 445–457

We present the definition of ρ-perturbations of an abstract wave equation. As a special case, this definition involves perturbations with compact support for the classical wave equation. We construct the scattering matrix for equations of such a type.

### On elements of the Lax-Phillips scattering scheme for $ρ$-perturbations of an abstract wave equation

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1615–1629

We give the definition of $ρ$-perturbations of an abstract wave equation. As a special case, this definition includes perturbations with compact support for the classical wave equation. By using the Lax-Phillips method, we study scattering of “$ρ$-perturbed” systems and establish some properties of corresponding scattering matrices.

### Abstract Lax-Phillips scattering scheme for second-order operator-differential equations

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 452-463

We construct an analog of the Lax-Phillips scattering scheme for an abstract operator-differential equation *u* _{u}=-*Lu* under certain restrictions imposed on the operator *L*. In particular, we construct the incoming and outcoming subspaces and describe singularities of the scattering matrix in terms of the spaces of boundary values.

### Spectral analysis of doubly $J$-nonexpanding operators

Ukr. Mat. Zh. - 1993. - 45, № 3. - pp. 384–388

For a contracting operator in a space with an indefinite metric (i.e., for a doubly*J*-nonexpanding operator) a characteristic operator-function is defined. On the basis of a detailed investigation of the properties of regular dilatations and characteristic functions of doubly $J$-nonexpanding operators, a spectral analysis of these operators is carried out.

### Spaces of boundary values and regular extensions of Hermitian operators

Ukr. Mat. Zh. - 1990. - 42, № 6. - pp. 854–858