Publications (11)3.77 Total impact
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ABSTRACT: The paper is concerned with the completeness problem of root functions of general boundary value problems for first order systems of ordinary differential equations. We introduce and investigate the class of weakly regular boundary conditions. We show that this class is much broader than the class of regular boundary conditions introduced by G.D. Birkhoff and R.E. Langer. Our main result states that the system of root functions of a boundary value problem is complete and minimal provided that the boundary conditions are weakly regular. Moreover, we show that in some cases the weak regularity of boundary conditions is also necessary for the completeness. Also we investigate the completeness for 2×22×2 Dirac type equations subject to irregular boundary conditions. Emphasize that our results are the first results on the completeness for general first order systems even in the case of regular boundary conditions.Journal of Functional Analysis 10/2012; 263(7):1939–1980. · 1.25 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The paper is devoted to general 2×2 first order systems of ordinary differential equations with general linear boundary conditions. While the problem of completeness of root vectors has been studied thoroughly for a single higher order equation, for systems of differential equations much less is known. However, in several recent papers by the authors, the class of systems and boundary conditions, for which the set of root vectors is known to be complete or incomplete is being gradually extended. See, in particular, M. M. Malamud and L. L. Oridoroga [Funct. Anal. Appl. 34, No. 4, 308310 (2000); translation from Funkts. Anal. Prilozh. 34, No. 4, 88–90 (2000; Zbl 0979.34058)], M. M. Malamud and L. L. Oridoroga [Dokl. Math. 82, No. 3, 899–904 (2010); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 435, No. 3, 298–304 (2010; Zbl 1231.34152)]. In the paper under review, this process is continued. The authors study various classes of systems not covered by earlier results. Here, the completeness or incompleteness depend on lower order terms. In the case of a holomorphic potential matrix and some nondegeneracy assumptions, even necessary and sufficient conditions are obtained.Methods of Functional Analysis and Topology. 01/2012; 18(1).  [Show abstract] [Hide abstract]
ABSTRACT: The paper is concerned with the completeness problem of root functions of general boundary value problems for first order systems of ordinary differential equations. Namely, we introduce and investigate the class of \emph{weakly regular boundary conditions}. We show that this class is much broader than the class of {\em regular boundary conditions} introduced by G.D. Birkhoff and R.E. Langer. Our main result states that the system of root functions of a boundary value problem is complete and minimal provided that the boundary conditions are weakly regular. Moreover, we show that in some cases \emph{the weak regularity} of boundary conditions \emph{is also necessary} for the completeness. Also we investigate the completeness for $2\times 2$ Dirac and Dirac type equations subject to irregular or even to degenerate boundary conditions. We emphasize that our results are the first results on the completeness problem for general first order systems even in the case of regular boundary conditions.Doklady Mathematics 09/2011; · 0.38 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We investigate negative spectra of 1D Schr\"odinger operators with $\delta$ and $\delta'$interactions on a discrete set in the framework of a new approach. Namely, using technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results of S. Albeverio and L. Nizhnik. For instance, we propose the algorithm for determining the number of negative squares of the operator with $\delta$interactions. We also show that the number of negative squares of the operator with $\delta'$interactions equals the number of negative strengths. Comment: 14 pages03/2009; 
Article: Theorem of Completeness for a DiracType Operator with Generalized λDepending Boundary Conditions
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ABSTRACT: . A completeness theorem is proved involving a system of integrodifferential equations with some λdepending boundary conditions. Also some sufficient conditions for the root functions to form a Riesz basis are established.Integral Equations and Operator Theory 01/2009; 64(3):357379. · 0.71 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: A completeness theorem is proved involving a system of integrodifferential equations with some $\lambda$depending boundary conditions. Also some sufficient conditions for the root functions to form a Riesz basis are established.08/2008;  Mathematical Notes 07/2008; 84(1):125129. · 0.24 Impact Factor
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ABSTRACT: The authors look at the first order differential equation iBy ' +Qy=λy, where y is the vector y=(y 1 ,⋯,y n ) T ,y i ∈L 2 (0,1), B=B * =diag(b 1 ,b 2 ,⋯,b n ) and Q =(q ij ) 1≤i,j≤n where q ij ∈L ∞ (0,1)· The boundary conditions depend on the eigenvalue parameter P 0 (λ)y(0)+P 1 (λ)y(1)=0, where P i is a matrix polynomial such that det(P 0 P 0 * +P 1 P 1 * )>0 for λ∈ℂ· Under the above conditions the authors show that the eigenfunctions form a complete system in ⨁ 1 n L 2 (0,1)· Furthermore, if we remove certain “extra” associated eigenfunctions, then the remaining set is a Riesz basis.Mathematical Notes 02/2006; 79(3):589593. · 0.24 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Pi0(λ)y 2 (0) + Pi1(λ)y � 2 (0) + Pi2(λ)y 2 1 + Pi3(λ)y � 2 1 + Pi4(λ)y(0)y � (0) + Pi5(λ)y(0)y 1 + Pi6(λ)y(0)y � 1 + Pi7(λ)y � (0)y 1Mathematical Notes 06/2003; 74(1):302307. · 0.24 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We consider the secondorder linear differential equation y" + A(t)y = 0 on the semiaxis with complexvalued potential function. Sufficient conditions for the potential function assuring that all solutions of the equation converge to zero at infinity are obtained. It is shown that the conditions imposed on the potential function are close to the necessary ones. One of the results seems to be new even in the case of realvalued function A(·).Journal of Mathematical Sciences 182(1).  [Show abstract] [Hide abstract]
ABSTRACT: We investigate negative spectra of onedimensional (1D) Schrödinger operators with δ and δ′interactions on a discrete set in the framework of a new approach. Namely, using the technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results of Albeverio and Nizhnik (Lett Math Phys 65:27–35, 2003; Methods Funct Anal Topol 9(4):273–286, 2003). For instance, we propose an algorithm for determining the number of negative squares of the operator with δinteractions. We also show that the number of negative squares of the operator with δ′interactions equals the number of negative strengths. Mathematics Subject Classification (2000)Primary 47A10–Secondary 34L40Integral Equations and Operator Theory 67(1):114. · 0.71 Impact Factor
Publication Stats
38  Citations  
3.77  Total Impact Points  
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Institutions

2003–2012

Donetsk National Medical University
Yuzovo, Donets’ka Oblast’, Ukraine
