Bessel Inequality and the Basis Property for a $$2m\times 2m $$ Dirac Type System with an Integrable Potential

Kurbanov, V. M.; Gadzhieva, G. R.

doi:10.1134/s0012266120050031

Cited by 7 publications

(6 citation statements)

References 14 publications

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“…(iii) In [24,25], Bessel and Riesz basis properties on abstract level were established, i.e. the operator L U (Q) was studied without explicit boundary conditions.…”

Section: Uniform Minimality and Riesz Basis Propertymentioning

confidence: 99%

“…Treated boundary conditions form rather broad class that covers, in particular, periodic, antiperiodic, and regular separated (not necessarily self-adjoint) boundary conditions. Note also that BVP for 2m × 2m Dirac equation (B = diag(−I m , I m )) was investigated in [51] (Bari-Markus property for Dirichlet BVP with Q ∈ L 2 ([0, 1]; C 2m×2m ) and in [24,25] (Bessel and Riesz basis properties on abstract level).…”

Section: Introductionmentioning

confidence: 99%

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On transformation operators and Riesz basis property of root vectors system for $n \times n$ Dirac type operators. Application to the Timoshenko beam model

Lunyov¹,

Malamud²

2021

Preprint

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The paper is concerned with the following n × n Dirac type equationWe show the existence of triangular transformation operators for such equation under additional uniform separation conditions on the entries of the matrix function B. Here we apply this result to study direct spectral properties of the boundary value problem (BVP) associated with the above equation subject to the general boundary conditions U(y) = Cy(0) + Dy(ℓ) = 0, rank(C D) = n.As a first application of this result, we show that the deviation of the characteristic determinants of this BVP and the unperturbed BVP (with Q = 0) is a Fourier transform of some summable function explicitly expressed via kernels of the transformation operators. In turn, this representation yields asymptotic behavior of the spectrum in the case of regular boundary conditions. Namely, λ m = λ 0 m + o(1) as m → ∞, where {λ m } m∈Z and {λ 0 m } m∈Z are sequences of eigenvalues of perturbed and unperturbed (Q = 0) BVP, respectively.Further, we prove that the system of root vectors of the above BVP constitutes a Riesz basis in a certain weighted L 2 -space, provided that the boundary conditions are strictly regular. Along the way, we also establish completeness, uniform minimality and asymptotic behavior of root vectors.The main results are applied to establish asymptotic behavior of eigenvalues and eigenvectors, and the Riesz basis property for the dynamic generator of spatially non-homogenous damped Timoshenko beam model. We also found a new case when eigenvalues have an explicit asymptotic, which to the best of our knowledge is new even in the case of constant parameters of the model.

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“…(iii) In [24,25], Bessel and Riesz basis properties on abstract level were established, i.e. the operator L U (Q) was studied without explicit boundary conditions.…”

Section: Uniform Minimality and Riesz Basis Propertymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On transformation operators and Riesz basis property of root vectors system for $n \times n$ Dirac type operators. Application to the Timoshenko beam model

Lunyov¹,

Malamud²

2021

Preprint

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“…Malamud also established the Riesz basis property with parentheses of the system of root vectors for different classes of BVPs for the 𝑛 × 𝑛 system with arbitrary 𝐵 of the form (1.5) and 𝑄 ∈ 𝐿 ∞ ([0, 1]; ℂ 𝑛×𝑛 ). Note also that BVP for the 2𝑚 × 2𝑚 Dirac equation (𝐵 = diag(−𝐼 𝑚 , 𝐼 𝑚 )) was investigated in [39] (Bari-Markus property for Dirichlet BVP with 𝑄 ∈ 𝐿 2 ([0, 1]; ℂ 2𝑚×2𝑚 ) and in [22,23] (Bessel and Riesz basis properties on abstract level).…”

Section: Introductionmentioning

confidence: 99%

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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“…Malamud also established the Riesz basis property with parentheses of the system of root vectors for different classes of BVPs for n × n system with arbitrary B of the form (1.5) and Q ∈ L ∞ ([0, 1]; C n×n ). Note also that BVP for 2m × 2m Dirac equation (B = diag(−I m , I m )) were investigated in [37] (Bari-Markus property for Dirichlet BVP with Q ∈ L 2 ([0, 1]; C 2m×2m ) and in [21,22] (Bessel and Riesz basis properties on abstract level).…”

Section: Introductionmentioning

confidence: 99%

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

Lunyov¹

2022

Preprint

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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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Bessel Inequality and the Basis Property for a $$2m\times 2m $$ Dirac Type System with an Integrable Potential

Cited by 7 publications

References 14 publications

On transformation operators and Riesz basis property of root vectors system for $n \times n$ Dirac type operators. Application to the Timoshenko beam model

On transformation operators and Riesz basis property of root vectors system for $n \times n$ Dirac type operators. Application to the Timoshenko beam model

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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