On the Riesz basis property of the root vector system for Dirac-type 2 × 2 systems

Lunyov, Anton A.; Malamud, M. M.

doi:10.1134/s106456241406012x

Cited by 17 publications

(37 citation statements)

References 10 publications

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“…The main results of the paper including Theorem 1.1 were announced in [28] (partially with proofs). After appearance of [28] there appeared the paper by A.M. Savchuk and A.A. Shkalikov [44] where Theorem 1.1 was proved for the 2 × 2 Dirac operator. Note that approaches in [28] and [44] substantially differ.…”

Section: References 38mentioning

confidence: 99%

On the Riesz basis property of root vectors system for 2 × 2 Dirac type operators

Lunyov

Malamud

2016

Journal of Mathematical Analysis and Applications

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The paper is concerned with the Riesz basis property of a boundary value problem associated in L 2 [0, 1] ⊗ C 2 with the following 2 × 2 Dirac type equationIt is proved that the system of root functions of a linear boundary value problem constitutes a Riesz basis in L 2 [0, 1] ⊗ C 2 provided that the boundary conditions are strictly regular. By analogy with the case of ordinary differential equations, boundary conditions are called strictly regular if the eigenvalues of the corresponding unperturbed (Q = 0) operator are asymptotically simple and separated. As distinguished from the Dirac case there is no simple algebraic criterion of the strict regularity whenever b1 + b2 = 0. However under certain restrictions on coefficients of the boundary linear forms we present certain algebraic criteria of the strict regularity in the latter case. In particular, it is shown that regular separated boundary conditions are always strictly regular while periodic (antiperiodic) boundary conditions are strictly regular if and only if b1 + b2 = 0.The proof of the main result is based on existence of triangular transformation operators for system (0.1). Their existence is also established here in the case of a summable Q. In the case of regular (but not strictly regular) boundary conditions we prove the Riesz basis property with parentheses. The main results are applied to establish the Riesz basis property of the dynamic generator of spatially non-homogenous damped Timoshenko beam model.2010 Mathematics Subject Classification. 47E05, 34L40, 34L10, 35L35.

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Section: References 38mentioning

confidence: 99%

On the Riesz basis property of root vectors system for 2 × 2 Dirac type operators

Lunyov

Malamud

2016

Journal of Mathematical Analysis and Applications

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“…uniformly in t ∈ [0, 1], which implies the asymptotic expansion in (6.6) if we note that l j t = x− v. Further, it is not difficult to check that the unperturbed eigenvalues λ 0 k in [LM14a,(32)] can be computed explicitly to be (note that b 1 − b 2 = −2l j therein) λ 0 k = −µ 0 j,k , k ∈ Z. Now the asymptotic expansion (6.7) follows from [LM14a,(33)].…”

Section: 1mentioning

confidence: 90%

“…As a first tool, we invoke results on the asymptotics of the eigenvalues of Dirac-Krein operators on a compact interval with summable potentials (see e.g. [AHM05], [LM14a]). In view of the boundary conditions (3.2) imposed for the operators T j on an edge e j = [v, v j ] for fixed j ∈ {1, 2, .…”

Section: 1mentioning

confidence: 99%

“…. , n}, we need to apply the more general results in [LM14a] and the detailed version [LM15] (see also [LM14b]). To this end, we first have to transform the differential equation (3.1) into the form considered in [LM14a] by means of the unitary matrix W diagonalizing J,…”

Section: 1mentioning

confidence: 99%

“…Proof. If we transform the independent variable x ∈ e j = [v, v j ] to t ∈ [0, 1] by means of x = v + l j t, we find that the system (6.2) is a special case of the more general Dirac-type system [LM14a,(1)…”

Section: 1mentioning

confidence: 99%

See 2 more Smart Citations

Dirac–Krein Systems on Star Graphs

Adamyan

Langer

Tretter

et al. 2016

Integr. Equ. Oper. Theory

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Abstract. We study the spectrum of a self-adjoint Dirac-Krein operator with potential on a compact star graph G with a finite number n of edges. This operator is defined by a Dirac-Krein differential expression with summable matrix potentials on each edge, by self-adjoint boundary conditions at the outer vertices, and by a self-adjoint matching condition at the common central vertex of G. Special attention is paid to Robin matching conditions with parameter τ ∈ R ∪ {∞}. Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference operator, we derive Krein's resolvent formula, introduce corresponding Weyl-Titchmarsh functions, study the multiplicities, dependence on τ , and interlacing properties of the eigenvalues, and prove a trace formula. Moreover, we show that, asymptotically for R → ∞, the difference of the number of eigenvalues in the intervals [0, R) and [−R, 0) deviates from some integer κ0, which we call dislocation index, at most by n + 2.Mathematics Subject Classification (2010). 81Q10, 81Q35, 47A10, 47A75.

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On convergence of spectral expansions of Dirac operators with regular boundary conditions

Макин¹

2022

Mathematische Nachrichten

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On the Riesz basis property of the root vector system for Dirac-type 2 × 2 systems

Cited by 17 publications

References 10 publications

On the Riesz basis property of root vectors system for 2 × 2 Dirac type operators

On the Riesz basis property of root vectors system for 2 × 2 Dirac type operators

Dirac–Krein Systems on Star Graphs

On convergence of spectral expansions of Dirac operators with regular boundary conditions

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