Leonid Nizhnik
@kiev.ua
Department of Functional Analysis
Institute of Mathematics NAS of Ukraine
RESEARCH, TEACHING, or OTHER INTERESTS
Mathematics, Analysis, Applied Mathematics
Scopus Publications
Scopus Publications
L. P. Nizhnik and I. L. Nizhnik
Springer Science and Business Media LLC
Kamila Dębowska and Leonid P. Nizhnik
AGHU University of Science and Technology Press
The main purposes of this paper are to study the direct and inverse spectral problems of the one-dimensional Dirac operators with nonlocal potentials. Based on informations about the spectrum of the operator, we find the potential and recover the form of the Dirac system. The methods used allow us to reduce the situation to the one-dimensional case. In accordance with the given assumptions and conditions we consider problems in a specific way. We describe the spectrum, the resolvent, the characteristic function etc. Illustrative examples are also given.
Sergii Kuzhel and Leonid Nizhnik
Springer Science and Business Media LLC
Let $S$ be a symmetric operator with equal defect numbers and let $\\mathfrak{U}$ be a set of unitary operators in a Hilbert space $\\mathfrak{H}$. The operator $S$ is called $\\mathfrak{U}$-invariant if $US=SU$ for all $U\\in\\mathfrak{U}$. Phillips \\cite{PH} constructed an example of $\\mathfrak{U}$-invariant symmetric operator $S$ which has no $\\mathfrak{U}$-invariant self-adjoint extensions. It was discovered that such symmetric operator has a constant characteristic function \\cite{KO}. For this reason, each symmetric operator $S$ with constant characteristic function is called a \\emph{Phillips symmetric operator}.
P.A. Cojuhari and L.P. Nizhnik
Elsevier BV
Johannes F. Brasche and Leonid Nizhnik
Element d.o.o.
We give an abstract definition of a one-dimensional Schr\\"odinger
operator with $\\delta'$-interaction on an arbitrary set~$\\Gamma$ of
Lebesgue measure zero. The number of negative eigenvalues of such an
operator is at least as large as the number of those isolated points
of the set~$\\Gamma$ that have negative values of the intensity
constants of the $\\delta'$-interaction. In the case where the
set~$\\Gamma$ is endowed with a Radon measure, we give constructive
examples of such operators having an infinite number of negative eigenvalues.
Leonid Nizhnik
Tamkang Journal of Mathematics
Leonid Nizhnik
IOP Publishing
We solve direct and inverse spectral problems for Sturm–Liouville operators with singular nonlocal potentials and nonlocal boundary conditions.
Sergio ALBEVERIO, Sergei KUZHEL, and Leonid P. NIZHNIK
Tokyo Journal of Mathematics
. We show that all types of self-adjoint perturbations of a semi-bounded operator A (purely singular, mixed singular, and regular) can be described and studied from a unique point of view in the framework of the extension theory as well as in the framework of the additive perturbation theory. We also show that any singular finite rank perturbation (cid:1) A can be approximated in the norm resolvent sense by regular finite rank perturbations of A . An application is given to the study of Schrödinger operators with point interactions.
Sergio Albeverio and Leonid Nizhnik
Elsevier BV
S. Albeverio, S. Kuzhel’, and L. Nizhnik
Springer Science and Business Media LLC
S Albeverio, R O Hryniv, and L P Nizhnik
IOP Publishing
We solve the inverse spectral problem for a class of Sturm–Liouville operators with singular non-local potentials and non-local boundary conditions. We study to what extent the operator from that class is determined by its spectrum, and point out subclasses in which the reconstruction problem from one spectrum has a unique solution.
Sergio Albeverio and Leonid Nizhnik
Wiley
AbstractA construction of a one‐dimensional Schrödinger operator that has an inner structure defined on a set of Lebesgue measure zero and an interaction given on such a set. General Krein–Feller operators are constructed and the spectrum of a Schrödinger operator with a δ′‐interaction given on a Cantor set is studied. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
L. P. Nizhnik
Springer Science and Business Media LLC
A. V. Kosyak and L. P. Nizhnik
Springer Science and Business Media LLC
S. Albeverio, G. Galperin, I.L. Nizhnik, and L.P. Nizhnik
Pleiades Publishing Ltd
Sergio Albeverio, Leonid Nizhnik, and Victor Tarasov
IOP Publishing
The existence of the scattering operator for a Dirac system with nonstationary point interactions is proved. A solution of the inverse scattering problem on determining the position and the characteristics of the point interactions from a given scattering operator is also presented.
Sergio Albeverio and Leonid Nizhnik
Springer Science and Business Media LLC
Sergio Albeverio and Leonid Nizhnik
Springer Science and Business Media LLC
S. Albeverio, V. Koshmanenko, P. Kurasov, and L. Nizhnik
American Mathematical Society (AMS)
Approximations of rank one H − 2 {\\mathcal H}_{-2} -perturbations of self-adjoint operators by operators with regular rank one perturbations are discussed. It is proven that in the case of arbitrary not semibounded operators such approximations in the norm resolvent sense can be constructed without any renormalization of the coupling constant. Approximations of semibounded operators are constructed using rank one non-symmetric regular perturbations.
L. P. Nizhnik
Springer Science and Business Media LLC
LEONID P. NIZHNIK, IRINA L. NIZHNIK, and MARTIN HASLER
World Scientific Pub Co Pte Lt
In this paper we present the construction of stable stationary solutions in reaction–diffusion systems consisting of a 1-D array of bistable cells with a cubic nonlinearity and with a cubic-like piecewise-linear nonlinearity. Some periodic solutions, kinks, solitons are considered. While it is known that spatial chaos arises in such systems with small coupling constants, we show the existence of spatial chaos for an arbitrary value of the cell coupling constant, in the case of the piecewise-linear nonlinearity. The value of the spatial entropy is found. We also show the existence of stable spatially periodic (pattern) solutions that persist for large coupling constants.
L. P. Nizhnik and V. G. Tarasov
Springer Science and Business Media LLC