# L. P. Nizhnik

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### Inverse spectral problems for Jacobi matrix with finite perturbed parameters

Methods Funct. Anal. Topology **21** (2015), no. 3, 256-265

For Jacobi matrices with finitely perturbed parameters, we get an explicit re\-presentation of the Weyl function, and solve inverse spectral problems, that is, we recover Jacobi matrices from spectral data. For the spectral data, we take the following: the spectral density of the absolutely continuous spectrum, with or without all the eigenvalues; the numerical parameters of the representation of one component of the vector-eigenfunction in terms of Chebyshev polynomials. We prove that these inverse problems have a unique solution, or only a finite number of solutions.

### Spectral analysis of metric graphs with infinite rays

Methods Funct. Anal. Topology **20** (2014), no. 4, 391-396

We conduct a detailed analysis for finite metric graphs that have a semi-infinite chain (a ray) attached to each vertex. We show that the adjacency matrix of such a graph gives rise to a selfadjoint operator that is unitary equivalent to a direct sum of a finite number of simplest Jacobi matrices. This permitted to describe spectrums of such operators and to explicitly construct an eigenvector decomposition.

### Schrödinger operators with nonlocal potentials

Sergio Albeverio, Leonid Nizhnik

Methods Funct. Anal. Topology **19** (2013), no. 3, 199-210

We describe selfadjoint nonlocal boundary-value conditions for new exact solvable models of Schrödinger operators with nonlocal potentials. We also solve the direct and the inverse spectral problems on a bounded line segment, as well as the scattering problem on the whole axis for first order operators with a nonlocal potential.

### One-dimensional Schrödinger operators with general point interactions

Methods Funct. Anal. Topology **19** (2013), no. 1, 4-15

We consider various forms of boundary-value conditions for general one-dimensional Schrödinger operators with point interactions that include $\delta$-- and $\delta'$-- interactions, $\delta'$-- potential, and $\delta$-- magnetic potential. We give most simple spectral properties of such operators, and consider a possibility of finding their norm resolvent approximations.

### Inverse eigenvalue problems for nonlocal Sturm-Liouville operators on a star graph

Methods Funct. Anal. Topology **18** (2012), no. 1, 68-78

We solve the inverse spectral problem for a class of Sturm--Liouville operators with singular nonlocal potentials and nonlocal boundary conditions on a star graph.

### On generalized selfadjoint operators on scales of Hilbert spaces

Yu. M. Berezansky, J. Brasche, L. P. Nizhnik

Methods Funct. Anal. Topology **17** (2011), no. 3, 193-198

We consider examples of generalized selfadjoint operators that act from a positive Hilbert space to a negative space. Such operators were introduced and studied in [1]. We give examples of selfadjoint operators on the principal Hilbert space $H_ 0$ that, being considered as operators from the positive space $H_ + \subset H_ 0$ into the negative space $H_ - \supset H_ 0$, are not essentially selfadjoint in the generalized sense.

### Singularly perturbed normal operators

Methods Funct. Anal. Topology **16** (2010), no. 4, 298-303

We give an effective description of finite rank singular perturbations of a normal operator by using the concepts we introduce of an admissible subspace and corresponding admissible operators. We give a description of rank one singular perturbations in terms of a scale of Hilbert spaces, which is constructed from the unperturbed operator.

### Inverse eigenvalue problems for nonlocal Sturm-Liouville operators

Methods Funct. Anal. Topology **15** (2009), no. 1, 41-47

We solve the inverse spectral problem for a class of Sturm-Liouville operators with singular nonlocal potentials and nonlocal boundary conditions.

### 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.

### Schrödinger operators with a number of negative eigenvalues equal to the number of point interactions

Sergio Albeverio, Leonid Nizhnik

Methods Funct. Anal. Topology **9** (2003), no. 4, 273-286

### A generalized sum of the quadratic forms

Methods Funct. Anal. Topology **8** (2002), no. 3, 13-19

### On rank one singular perturbations of selfadjoint operators

Methods Funct. Anal. Topology **7** (2001), no. 3, 54-66

### Boundary value problems with singular conditions on boundary components of small dimensions

Methods Funct. Anal. Topology **7** (2001), no. 1, 76-81

### Direct and inverse scattering problems for multidimensional system of first-order partial differential equations

Methods Funct. Anal. Topology **5** (1999), no. 2, 44-82

### Representations of double commutator by matrix-differential operators

Methods Funct. Anal. Topology **3** (1997), no. 3, 75-80