On the Riesz basis property of root vectors system for $2 \times 2$ Dirac type operators

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

doi:10.48550/arxiv.1504.04954

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“…Under a simple additional assumption this system forms a Bari basis if and only if BC are self-adjoint. In this paper we continue our investigation [31] of the following first order system of differential equations…”

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confidence: 99%

“…The Riesz basis property in L 2 ([0, 1]; C 2 ) of BVP (1.1)-(1.3), i.e. of the operator L U (Q), for 2 × 2 Dirac system with various assumptions on the potential matrix Q was investigated in numerous papers (see [52,53,41,42,19,8,3,11,10,9,12,28,29,48,31] and references therein). At that time the most strong result was obtained by P. Djakov and B. Mityagin [8,11] and A. Baskakov, A. Derbushev, A. Shcherbakov [3] who proved under the assumption Q ∈ L 2 ([0, 1]; C 2×2 ) that the root vectors system of the BVP (1.1)-(1.3) with strictly regular BCs forms a Riesz basis and forms a Riesz basis with parentheses whenever BC are only regular.…”

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“…Later the case Q ∈ L 1 ([0, 1]; C 2×2 ) was treated independently and with different methods by the authors [29,31], on the one hand, and by A.M. Savchuk and A.A. Shkalikov [48], on the other hand. Namely, it was proved that BVP (1.1)-(1.3) with Q ∈ L 1 ([0, 1]; C 2×2 ) and strictly regular boundary conditions has the Riesz basis property while BVP with only regular BC has the property of Riesz basis with parentheses.…”

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“…To describe our approach to Riesz basis property used in [29,31] let us denote by e ± (•, λ) solution to system (1.1) satisfying the initial conditions e ± (0, λ) = 1 ±1 . Our investigation in [29,31] substantially relies on the following representation of e ± (•, λ) by means of triangular transformation operators:…”

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“…relating the eigenvalues Λ = {λ n } n∈Z and Λ 0 = {λ 0 n } n∈Z of the operators L U (Q) and L U (0) (with regular BC ), respectively, (see [29,31] for details and also [48] where formula (1.9) was obtained by using different method). Note also that representation (1.8) for the determinant ∆ Q (•) was substantially used in a recent papers by A. Makin [33], [34].…”

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Stability of spectral characteristics and Bari basis property of boundary value problems for $2 \times 2$ Dirac type systems

Lunyov,

Malamud

2020

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The paper is concerned with the stability property under perturbation Q → Q of different spectral characteristics of a boundary value problem associated in L 2 ([0, 1]; C 2 ) with the following 2 × 2 Dirac type equation 2010 Mathematics Subject Classification. 47E05, 34L40, 34L10, 34L15. Key words and phrases. Dirac type systems; spectral stability; regular and strictly regular boundary conditions; transformation operators; Bari basis property. 1 2 ANTON A. LUNYOV AND MARK M. MALAMUD 5. Fourier transform estimates 28 5.1. Generalization of Hausdorff-Young and Hardy-Littlewood theorems 28 5.2. Uniform versions of Riemann-Lebesgue Lemma 33 6. Stability property of eigenvalues 36 6.1. Uniform localization of spectrum 36 6.2. Stability property of eigenvalues for Q ∈ L p 40 7. Stability property of eigenfunctions 45 7.1. Estimates of Fourier transforms of transformation operators 45 7.2. Stability of Fourier transforms of transformation operators 50 7.3. Stability property of the fundamental matrix 53 7.4. Stability property of the eigenfunctions 56 8.

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Stability of spectral characteristics and Bari basis property of boundary value problems for $2 \times 2$ Dirac type systems

Lunyov,

Malamud

2020

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Criterion of Bari basis property for 2 × 2 Dirac‐type operators with strictly regular boundary conditions

Lunyov

2023

Mathematische Nachrichten

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The paper is concerned with the Bari basis property of a boundary value problem associated in 𝐿 2 ([0, 1]; ℂ 2 ) with the following 2 × 2 Dirac-type equation for 𝑦 = col(𝑦 1 , 𝑦 2 ):with a potential matrix 𝑄 ∈ 𝐿 2 ([0, 1]; ℂ 2×2 ) and subject to the strictly regular boundary conditions 𝑈𝑦 ∶= {𝑈 1 , 𝑈 2 }𝑦 = 0. If 𝑏 2 = −𝑏 1 = 1, this equation is equivalent to one-dimensional Dirac equation. We show that the normalized system of root vectors {𝑓 𝑛 } 𝑛∈ℤ of the operator 𝐿 𝑈 (𝑄) is a Bari basis in 𝐿 2 ([0, 1]; ℂ 2 ) if and only if the unperturbed operator 𝐿 𝑈 (0) is self-adjoint. We also give explicit conditions for this in terms of coefficients in the boundary conditions. The Bari basis criterion is a consequence of our more general result: Let 𝑄 ∈ 𝐿 𝑝 ([0, 1]; ℂ 2×2 ), 𝑝 ∈ [1, 2], boundary conditions be strictly regular, and let {𝑔 𝑛 } 𝑛∈ℤ be the sequence biorthogonal to the normalized system of root vectors {𝑓 𝑛 } 𝑛∈ℤ of the operator 𝐿 𝑈 (𝑄). Then,These abstract results are applied to noncanonical initial-boundary value problem for a damped string equation.

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Criterion of Bari basis property for $2 \times 2$ Dirac-type operators with strictly regular boundary conditions

Lunyov¹

2022

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The paper is concerned with the Bari basis property of a boundary value problem associated in L 2 ([0, 1]; C 2 ) with the following 2 × 2 Dirac-type equation for y = col(y 1 , y 2 ):) and subject to the strictly regular boundary conditionsWe show that the system of root vectors {f n } n∈Z of the operator L U (Q) forms a Bari basis in L 2 ([0, 1]; C 2 ) if and only if the unperturbed operator L U (0) is self-adjoint. We also give explicit conditions for this in terms of coefficients in the boundary conditions. The Bari basis criterion is a consequence of our more general result:boundary conditions be strictly regular, and let {g n } n∈Z be the sequence biorthogonal to the system of root vectors {f n } n∈Z of the operator L U (Q). ThenThese abstract results are applied to non-canonical initial-boundary value problem for a damped string equation.

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On the Riesz basis property of root vectors system for $2 \times 2$ Dirac type operators

Cited by 3 publications

References 0 publications

Stability of spectral characteristics and Bari basis property of boundary value problems for $2 \times 2$ Dirac type systems

Stability of spectral characteristics and Bari basis property of boundary value problems for $2 \times 2$ Dirac type systems

Criterion of Bari basis property for 2 × 2 Dirac‐type operators with strictly regular boundary conditions

Criterion of Bari basis property for $2 \times 2$ Dirac-type operators with strictly regular boundary conditions

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