Leonid Nizhnik

RESEARCH, TEACHING, or OTHER INTERESTS

Mathematics, Analysis, Applied Mathematics

Scopus Publications

Deterministic Diffusion
L. P. Nizhnik and I. L. Nizhnik
Springer Science and Business Media LLC



Scattering problem for Dirac system with nonlocal potentials




Direct and inverse spectral problems for Dirac systems with nonlocal potentials
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.


Phillips symmetric operators and their extensions
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}.


On the inverse eigenvalue problems for a jacobi matrix with mixed given data




Hochstadt inverse eigenvalue problem for Jacobi matrices
P.A. Cojuhari and L.P. Nizhnik
Elsevier BV



On new inverse spectral problems for weighted graphs




One-dimensional schrödinger operators with δ'-interactions on a set of lebesgue measure zero
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.


Inverse spectral nonlocal problem for the first order ordinary differential equation
Leonid Nizhnik
Tamkang Journal of Mathematics



Inverse nonlocal Sturm-Liouville problem
Leonid Nizhnik
IOP Publishing
We solve direct and inverse spectral problems for Sturm–Liouville operators with singular nonlocal potentials and nonlocal boundary conditions.


On the Perturbation Theory of Self-Adjoint Operators
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.


Schrödinger operators with nonlocal point interactions
Sergio Albeverio and Leonid Nizhnik
Elsevier BV



Singularly perturbed self-adjoint operators in scales of Hilbert spaces
S. Albeverio, S. Kuzhel’, and L. Nizhnik
Springer Science and Business Media LLC



Inverse spectral problems for non-local Sturm-Liouville operators
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.


A Schrödinger operator with a δ′-interaction on a Cantor set and Krein-Feller operators
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)


A one-dimensional Schrödinger operator with point interactions on Sobolev spaces
L. P. Nizhnik
Springer Science and Business Media LLC



Operators of generalized translation and hypergroups constructed from self-adjoint differential operators
A. V. Kosyak and L. P. Nizhnik
Springer Science and Business Media LLC



Generalized billiards inside an infinite strip with periodic laws of reflection along the strip's boundaries
S. Albeverio, G. Galperin, I.L. Nizhnik, and L.P. Nizhnik
Pleiades Publishing Ltd



Inverse scattering problem for a Dirac system with nonstationary point interactions
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.


A schrödinger operator with point interactions on sobolev spaces
Sergio Albeverio and Leonid Nizhnik
Springer Science and Business Media LLC



On the number of negative eigenvalues of a one-dimensional Schrödinger operator with point interactions
Sergio Albeverio and Leonid Nizhnik
Springer Science and Business Media LLC



On approximations of rank one ℋ<inf>-2</inf>-perturbations
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.


A Schrödinger Operator with δ′-Interaction
L. P. Nizhnik
Springer Science and Business Media LLC



Stable stationary solutions in reaction-diffusion systems consisting of a 1-D array of bistable cells
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.


Scattering problem for a multidimensional system of first-order partial differential equations
L. P. Nizhnik and V. G. Tarasov
Springer Science and Business Media LLC

RECENT SCHOLAR PUBLICATIONS

• Well-posed Cauchy problem and the Hamiltonian form of (2+ 1) nonlinear equations integrable by inverse scattering transform
  L Nizhnik
  arXiv preprint arXiv:2412.16801 , 2024
  2024
  Citations: 1
• The Cauchy problem for the (2+ 1) integrable nonlinear Schr\" odinger equation"
  LP Nizhnik
  arXiv preprint arXiv:2305.06182 , 2023
  2023
• Self-adjoint operators with nonlocal interaction
  L NIZHNIK
  CONTROL AND OPTIMIZATION WITH INDUSTRIAL APPLICATIONS, 20 , 2020
  2020
  Citations: 1
• Детермінована дифузія
  ІЛ Нижник, ЛП Нижник
  Ukrains’ kyi Matematychnyi Zhurnal 71 (11), 1553-1569 , 2019
  2019
• Scattering problem for Dirac system with nonlocal potentials
  PA Cojuhari, LP Nizhnik
  Methods of Functional Analysis and Topology 25 (03), 211-218 , 2019
  2019
  Citations: 6
• Direct and inverse spectral problems for Dirac systems with nonlocal potentials
  K Dębowska, LP Nizhnik
  Opuscula Mathematica 39 (5), 645-673 , 2019
  2019
  Citations: 9
• Phillips symmetric operators and their extensions
  S Kuzhel, L Nizhnik
  2018
  Citations: 7
• On the inverse eigenvalue problems for a Jacobi matrix with mixed given data
  LP Nizhnik
  Methods of Functional Analysis and Topology 24 (02), 178-186 , 2018
  2018
  Citations: 2
• Hochstadt inverse eigenvalue problem for Jacobi matrices
  PA Cojuhari, LP Nizhnik
  Journal of Mathematical Analysis and Applications 455 (1), 439-451 , 2017
  2017
  Citations: 5
• On new inverse spectral problems for weighted graphs
  LP Nizhnik, VI Rabanovich
  Methods of Functional Analysis and Topology 23 (1), 66-75 , 2017
  2017
  Citations: 6
• Inverse spectral problems for Jacobi matrix with finite perturbed parameters
  LP Nizhnik
  Methods of Functional Analysis and Topology 21 (3), 256-265 , 2015
  2015
  Citations: 2
• Differential Equations with Bistable Nonlinearity
  ЛП Нижник, АМ Самойленко
  Ukrains’ kyi Matematychnyi Zhurnal 67 (4), 517-554 , 2015
  2015
• Deterministic Diffusion
  L Nizhnik, I Nizhnik
  arXiv preprint arXiv:1501.00674 , 2015
  2015
  Citations: 1
• Spectral analysis of metric graphs with infinite rays
  LP Nizhnik
  Methods of Functional Analysis and Topology 20 (4), 391-396 , 2014
  2014
  Citations: 7
• Спектральний аналіз деяких графів з нескінченними променями
  ВО Лебідь, ЛП Нижник
  Ukrains’ kyi Matematychnyi Zhurnal 66 (9), 1193-1204 , 2014
  2014
  Citations: 1
• Спектральний аналiз локально скiнченних графiв з одним нескiнченним променем
  ВО Лебiдь, ЛП Нижник
  Доповіді Національної академії наук України, 29-35 , 2014
  2014
  Citations: 2
• Спектральний аналіз локально скінченних графів з одним нескінченним променем
  ВО Лебідь, ЛП Нижник
  Видавничий дім" Академперіодика" НАН України , 2014
  2014
  Citations: 1
• Sergio Albeverio
  S Albeverio, GA Galperin, IL Nizhnik, LP Nizhnik
  2014
• Spectral analysis of locally finite graphs with one infinite ray
  VO Lebid, LP Nizhnik
  2014
  Citations: 7
• Юрій Стефанович Самойленко (до 70-річчя від дня народження)
  YM Berezansky, ОA Boichuk, ML Gorbachuk, YA Drozd, VS Korolyuk, ...
  Ukrains’ kyi Matematychnyi Zhurnal 65 (10), 1408-1409 , 2013
  2013

MOST CITED SCHOLAR PUBLICATIONS

• Integration of multidimensional nonlinear equations by the method of the inverse problem
  LP Nizhnik
  Doklady Akademii Nauk 254 (2), 332-335 , 1980
  1980
  Citations: 176
• Inverse scattering problems for hyperbolic equations
  LP Nizhnik
  Kiev, Nauk. Dumka Publ , 1991
  1991
  Citations: 115
• Inverse spectral problems for non-local Sturm–Liouville operators
  S Albeverio, RO Hryniv, LP Nizhnik
  Inverse problems 23 (2), 523-535 , 2007
  2007
  Citations: 102
• Inverse nonlocal Sturm-Liouville problem
  L Nizhnik
  Inverse problems 26 (12), 125006 , 2010
  2010
  Citations: 78
• Approximation of general zero-range potentials
  S Albeverio, L Nizhnik
  Ukrainian Mathematical Journal 52 (5), 664-672 , 2000
  2000
  Citations: 65
• Schrödinger operators with nonlocal point interactions
  S Albeverio, L Nizhnik
  Journal of mathematical analysis and applications 332 (2), 884-895 , 2007
  2007
  Citations: 56
• On the number of negative eigenvalues of a one-dimensional Schrödinger operator with point interactions
  S Albeverio, L Nizhnik
  Letters in Mathematical Physics 65 (1), 27-35 , 2003
  2003
  Citations: 51
• One-dimensional Schr\"odinger operators with -interactions on a set of Lebesgue measure zero"
  JF Brasche, L Nizhnik
  arXiv preprint arXiv:1112.2545 , 2011
  2011
  Citations: 48
• Inverse eigenvalue problems for nonlocal Sturm-Liouville operators
  LP Nizhnik
  Methods of Functional Analysis and Topology 15 (01), 41-47 , 2009
  2009
  Citations: 42
• Inverse spectral nonlocal problem for the first order ordinary differential equation
  L Nizhnik
  Tamkang Journal of Mathematics 42 (3), 385-394 , 2011
  2011
  Citations: 38
• Inverse eigenvalue problems for nonlocal Sturm-Liouville operators on a star graph
  LP Nizhnik
  Methods of Functional Analysis and Topology 18 (1), 68-78 , 2012
  2012
  Citations: 36
• The inverse scattering problems for the hyperbolic equations and their application to non-linear integrable systems
  LP Nizhnik
  Reports on mathematical physics 26 (2), 261-283 , 1988
  1988
  Citations: 36
• Integrability of nonlinear multidimensional equations with inverse scattering transform approach
  LP Nizhnik
  Doklady of AN USSR 254 (2), 332-335 , 1980
  1980
  Citations: 34
• The inverse nonstationary scattering problem
  LP Nizhnik
  Nauk. Dumka, Kiev , 1973
  1973
  Citations: 33
• Stable stationary solutions in reaction–diffusion systems consisting of a 1-d array of bistable cells
  LP Nizhnik, IL Nizhnik, M Hasler
  International Journal of Bifurcation and Chaos 12 (02), 261-279 , 2002
  2002
  Citations: 32
• On rank one singular perturbations of selfadjoint operators
  LP Nizhnik
  Methods of Functional Analysis and Topology 7 (3), 54-66 , 2001
  2001
  Citations: 32
• Integration of the nonlinear two-dimensional spatial Schrodinger equation by the inverse-problem method
  LP Nizhnik, MD Pochinaiko
  Functional Analysis and Its Applications 16 (1), 66-69 , 1982
  1982
  Citations: 29
• A Schrödinger operator with a δ ′‐interaction on a Cantor set and Krein–Feller operators
  S Albeverio, L Nizhnik
  Mathematische Nachrichten 279 (5‐6), 467-476 , 2006
  2006
  Citations: 28
• Inverse scattering problem on a semi-axis with a non-self-adjoint potential matrix
  LP Nizhnik, PL Vu
  Ukr. mat. zhurnal (Ukrainian Math. J.), 469-486 , 1974
  1974
  Citations: 28
• Schrödinger operators with a number of negative eigenvalues equal to the number of point interactions
  LP Nizhnik, S Albeverio
  Methods of Functional Analysis and Topology 9 (4), 273-286 , 2003
  2003
  Citations: 27