On the deficiency index of the vector-valued Sturm–Liouville operator

Mirzoev, K. A.; Safonova, T. A.

doi:10.1134/s0001434616010314

Cited by 12 publications

(10 citation statements)

References 10 publications

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“…19.2]. Condition (5) ensures the solvability of nonlinear equation (6). The form of this equation is directly related with the transformation operator + Γ * .…”

Section: Definition 1 Two Linear Operatorsmentioning

confidence: 99%

Asymptotics of eigenvalues of infinite block matrices

Braeutigam

Polyakov

2019

Ufimsk. Mat. Zh.

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The paper is devoted to determining the asymptotic behavior of eigenvalues, which is one of topical directions in studying operators generated by tridiagonal infinite block matrices in Hilbert spaces of infinite sequences with complex coordinates or, in other words, to discrete Sturm-Liouville operators. In the work we consider a class of non-selfadjoint operators with discrete spectrum being a sum of a self-adjoint operator serving as an unperturbed operator and a perturbation, which is an operator relatively compact with respect to the unperturbed operator. In order to study the asymptotic behavior of eigenvalues, in the paper we develop an adapted scheme of abstract method of similar operators. The main idea of this approach is that by means of the similarity operator, the studying of spectral properties of the original operator is reduced to studying the spectral properties of an operator of a simpler structure. Employing this scheme, we write out the formulae for the asymptotics of arithmetical means of the eigenvalues of the considered class of the operators. We note that such approach differs essentially from those employed before. The obtained general result is applied for determining eigenvalues of particular operators. Namely, we provide asymptotics for the eigenvalues of symmetric and non-symmetric tridiagonal infinite matrices in the scalar case, the asymptotics for arithmetical means of the eigenvalues of block matrices with power behavior of eigenvalues of unperturbed operator and generalized Jacobi matrices with various number of non-zero off-diagonals.

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“…19.2]. Condition (5) ensures the solvability of nonlinear equation (6). The form of this equation is directly related with the transformation operator + Γ * .…”

Section: Definition 1 Two Linear Operatorsmentioning

confidence: 99%

Asymptotics of eigenvalues of infinite block matrices

Braeutigam

Polyakov

2019

Ufimsk. Mat. Zh.

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“…Thus, we have verified the estimate The selfadjointness (but not discreteness) of the Hamiltonian H X,α was established in [35] for {d 2 n } ∞ 1 ∈ l 1 and in [10] under the condition (4.6), where it is removed from Theorem 2.5(ii). (ii) In the papers [32,33] for the case p = 1, and in [35,46] -for p ≥ 1 it is shown that the condition…”

Section: Discreteness Conditions For Jacobi Matricesmentioning

confidence: 99%

On the Deficiency Indices of Block Jacobi Matrices Related to Dirac Operators with Point Interactions

Budyka¹,

Malamud

2019

Math Notes

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The paper concerns with infinite block Jacobi matrices J with p × p-matrix entries. We present new conditions for these block matrices to be selfadjoint and have discrete spectrum. In our previous papers there was established a close relation between a class of such matrices and 2p × 2p Dirac operators with point interactions in L 2 (R; C 2p). For block Jacobi matrices J of this class we present several conditions ensuring either maximality or intermediatory of their deficiency indices. Applications to Dirac and matrix Schrodinger operators are given. It is worth mentioning that the above mentioned connection is employed here in both directions for the first time. In particular, the property of J to have maximal deficiency indices was firstly established for Dirac operators.

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“…Later this problem for classical matrix Sturm-Liouville operators and operators with generalized coefficients was discussed in many works, see, for instance, [3-5, 9, 11-13, 15-17, 19-22] (and also the references therein). In particular, for example, in [17] the authors obtained the conditions of nonmaximality of deficiency numbers of operator L 0 generated by (1.2). M.S.P.…”

Section: 3mentioning

confidence: 99%

“…Detailed justification of this fact is given in [17]. As above, using the matrix F , we can define the quasi-derivatives of given vector function y ∈ AC n,loc (I), assuming y [0] := y, y [1] := P 0 y − Φy, y [2] := (y [1] ) +ΦP −1 0 y [1] +ΦP −1 0 Φy.…”

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confidence: 99%

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Limit-point criteria for the matrix Sturm-Liouville operator and its powers

Braeutigam¹

2017

OpMath

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Abstract. We consider matrix Sturm-Liouville operators generated by the formal expression ). Let the matrix functions P := P (x), Q := Q(x) and R := R(x) of order n (n ∈ N) be defined on I, P is a nondegenerate matrix, P and Q are Hermitian matrices for x ∈ I and the entries of the matrix functions P −1 , Q and R are measurable on I and integrable on each of its closed finite subintervals. The main purpose of this paper is to find conditions on the matrices P , Q and R that ensure the realization of the limit-point case for the minimal closed symmetric operator generated by l k [y] (k ∈ N). In particular, we obtain limit-point conditions for Sturm-Liouville operators with matrix-valued distributional coefficients.

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On the deficiency index of the vector-valued Sturm–Liouville operator

Cited by 12 publications

References 10 publications

Asymptotics of eigenvalues of infinite block matrices

Asymptotics of eigenvalues of infinite block matrices

On the Deficiency Indices of Block Jacobi Matrices Related to Dirac Operators with Point Interactions

Limit-point criteria for the matrix Sturm-Liouville operator and its powers

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