Reconstruction for Sturm–Liouville operators with frozen argument for irrational cases

Wang, Yu Ping; Zhang, Maozhu; Zhao, Wenju; Wei, Xianbiao

doi:10.1016/j.aml.2020.106590

Cited by 19 publications

(25 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The above results for Sturm-Liouville-type operators with frozen argument are proved in several recent papers [1][2][3] using quite an involved technique that is based on the integral representation of the eigenfunctions v n and crucially depends on the special properties of the Sturm-Liouville differential expressions. In the context of the present research, these results are obtained as special cases of more general abstract statements proved in Sections 3 and 4, which demonstrates the efficiency of the proposed approach.…”

Section: Examples and Discussionmentioning

confidence: 88%

“…Thus we fix an operator A satisfying the standing assumptions (A1) and (A2), i.e., is self-adjoint and has a simple discrete spectrum (λ n ) n∈I that is d-separated as in (3). Assume further that ϕ is a vector in the Hilbert space H −α for some α < 2 such that (A3) holds.…”

Section: Inverse Spectral Problemmentioning

confidence: 99%

“…Bondarenko et al [2] extended the above research to the case of various boundary conditions. A more difficult case when a/π is irrational was recently discussed in [3]; uniqueness of q was proved and the reconstruction algorithm suggested. In [4], the authors considered the periodic boundary conditions for the differential expression (1); then the unperturbed operator with q ≡ 0 has eigenvalues of multiplicity 2, and the analysis of both direct and inverse spectral problems becomes technically more involved.…”

Section: Introductionmentioning

confidence: 99%

“…Since F is given by explicit formulae, this approach suggests a constructive algorithm of determining ψ from the spectrum of the perturbation B. After establishing these abstract results, we specialize them to a wide variety of differential operators with frozen arguments including those of Sturm-Liouville type (1) thus providing a unified treatment of the questions discussed in the papers [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Reconstruction of Differential Operators with Frozen Argument

Dobosevych

Hryniv

2022

Axioms

View full text Add to dashboard Cite

We study spectral properties of a wide class of differential operators with frozen arguments by putting them into a general framework of rank-one perturbation theory. In particular, we give a complete characterization of possible eigenvalues for these operators and solve the inverse spectral problem of reconstructing the perturbation from the resulting spectrum. This approach provides a unified treatment of several recent studies and gives a clear explanation and interpretation of the obtained results.

show abstract

Section: Examples and Discussionmentioning

confidence: 88%

Section: Inverse Spectral Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Reconstruction of Differential Operators with Frozen Argument

Dobosevych

Hryniv

2022

Axioms

View full text Add to dashboard Cite

show abstract

“…We note that some analogous as well as different models leading to ordinary functionaldifferential equations with one or several frozen arguments were given, e.g., in [10,11]. Various aspects of inverse problems for operators with frozen argument were studied in [13][14][15][16][17][18][19][20][21][22][23]. In particular, in [21] the problem B was considered for a ∈ [0, 1] ∩ Q, i.e.…”

Section: Introductionmentioning

confidence: 99%

On the inverse problem for Sturm–Liouville-type operators with frozen argument: rational case

Buterin

Kuznetsova

2019

Comp. Appl. Math.

View full text Add to dashboard Cite

We study two inverse problems of recovering the complex-valued squareintegrable potential q(x) in the operator −y ′′ (x) + q(x)y(a), y (ν) (0) = γy (ν) (1) for ν = 0, 1, where γ ∈ C \ {0}. The first problem involves only single spectrum as the input data. We obtain complete characterization of the spectrum and prove that its specification determines q(x) uniquely if and only if γ = ±1. For the rest cases: periodic and antiperiodic ones, we describe classes of iso-spectral potentials and provide restrictions under which the uniqueness holds. The second inverse problem deals with recovering q(x) from the two spectra related to γ = ±1. We obtain necessary and sufficient conditions for its solvability and establish that uniqueness holds if and only if a = 0, 1. For a ∈ (0, 1), we describe classes of iso-bispectral potentials and provide restrictions under which the uniqueness resumes. Algorithms for solving both inverse problems are provided.

show abstract

Sturm–Liouville‐type operators with frozen argument and Chebyshev polynomials

Tsai

Liu

Buterin

et al. 2022

Math Methods in App Sciences

View full text Add to dashboard Cite

In recent years, there appeared a growing interest in the inverse spectral theory for functional-differential operators with frozen argument. Such operators are nonlocal and belong to the so-called loaded differential operators, which frequently appear in mathematics as well as natural sciences and engineering. However, to various classes of nonlocal operators, classical methods of the inverse spectral theory are not applicable. For this reason, it is relevant to develop new methods and approaches for solving inverse problems for operators of this type. We establish a deep connection between the inverse problem for operators with frozen argument and Chebyshev polynomials of the first and the second kinds. Appearing to be of an interest from the point of view of the Chebyshev polynomials theory itself, this connection gives a new perspective method for studying inverse problems for operators with frozen argument. In particular, it allows one to completely describe all non-degenerate and degenerate cases, that is, when the corresponding inverse problem is uniquely solvable or not, respectively. Moreover, it gives a convenient description of all isospectral potentials in the degenerate case, which is demonstrated by some illustrative examples.

show abstract

Reconstruction for Sturm–Liouville operators with frozen argument for irrational cases

Cited by 19 publications

References 16 publications

Reconstruction of Differential Operators with Frozen Argument

Reconstruction of Differential Operators with Frozen Argument

On the inverse problem for Sturm–Liouville-type operators with frozen argument: rational case

Sturm–Liouville‐type operators with frozen argument and Chebyshev polynomials

Contact Info

Product

Resources

About