A representation of the transmutation kernels for the Schrödinger operator in terms of eigenfunctions and applications

Khmelnytskaya, Kira V.; Kravchenko, Vladislav V.; Torba, Sergii M.

doi:10.1016/j.amc.2019.02.024

Cited by 5 publications

(10 citation statements)

References 10 publications

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“…In the present work we improve the approach from [17,18] by using the representation for the Gelfand-Levitan kernel F(x, t) obtained recently in [16]. We prove the stability of the method and extend it onto the inverse Sturm-Liouville problem by two spectra (Problem 2, see its statement below as Problem 2.2).…”

Section: Introductionmentioning

confidence: 92%

A direct method for solving inverse Sturm–Liouville problems*

Kravchenko¹,

Torba²

2020

Inverse Problems

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We consider two main inverse Sturm–Liouville problems: the problem of recovery of the potential and the boundary conditions from two spectra or from a spectral density function. A simple method for practical solution of such problems is developed, based on the transmutation operator approach, new Neumann series of Bessel functions representations for solutions and the Gelfand–Levitan equation. The method allows one to reduce the inverse Sturm–Liouville problem directly to a system of linear algebraic equations, such that the potential is recovered from the first element of the solution vector. We prove the stability of the method and show its numerical efficiency with several numerical examples.

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Section: Introductionmentioning

confidence: 92%

A direct method for solving inverse Sturm–Liouville problems*

Kravchenko¹,

Torba²

2020

Inverse Problems

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“…The first coefficient of the Fourier-Legendre series which corresponds to the first component of the solution vector of the linear algebraic system is sufficient for recovering the Sturm-Liouville problem. In the present work we improve the approach from [17], [18] by using the representation for the Gelfand-Levitan kernel F (x, t) obtained recently in [16]. We prove the stability of the method and extend it onto the inverse Sturm-Liouville problem by two spectra (Problem 2).…”

Section: Introductionmentioning

confidence: 92%

“…Due to a slow convergence of the series (2.12) and the presence of a jump discontinuity of the series (2.12) at x = t = π (we refer to [16] for more details), it was proposed in [18,Section 13.4] to work with the integrated version of the Gelfand-Levitan equation…”

Section: The Transmutation Integral Kernel and The Gelfand-levitan Eq...mentioning

confidence: 99%

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A direct method for solving inverse Sturm-Liouville problems

Kravchenko,

Torba

2020

Preprint

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We consider two main inverse Sturm-Liouville problems: the problem of recovery of the potential and the boundary conditions from two spectra or from a spectral density function. A simple method for practical solution of such problems is developed, based on the transmutation operator approach, new Neumann series of Bessel functions representations for solutions and the Gelfand-Levitan equation. The method allows one to reduce the inverse Sturm-Liouville problem directly to a system of linear algebraic equations, such that the potential is recovered from the first element of the solution vector. We prove the stability of the method and show its numerical efficiency with several numerical examples.

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“…The series (2.37) converges slowly and possesses a jump discontinuity (for ω = 0) at x = t = π. To overcome these difficulties, in [19] another representation for the function F was derived. Namely,…”

Section: The Gelfand-levitan Equationmentioning

confidence: 99%

A practical method for recovering Sturm-Liouville problems from the Weyl function

Kravchenko,

Torba

2021

Preprint

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In the paper we propose a direct method for recovering the Sturm-Liouville potential from the Weyl-Titchmarsh m-function given on a countable set of points. We show that using the Fourier-Legendre series expansion of the transmutation operator integral kernel the problem reduces to an infinite linear system of equations, which is uniquely solvable if so is the original problem. The solution of this linear system allows one to reconstruct the characteristic determinant and hence to obtain the eigenvalues as its zeros and to compute the corresponding norming constants. As a result, the original inverse problem is transformed to an inverse problem with a given spectral density function, for which the direct method of solution from [24] is applied.The proposed method leads to an efficient numerical algorithm for solving a variety of inverse problems. In particular, the problems in which two spectra or some parts of three or more spectra are given, the problems in which the eigenvalues depend on a variable boundary parameter (including spectral parameter dependent boundary conditions), problems with a partially known potential and partial inverse problems on quantum graphs.

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A representation of the transmutation kernels for the Schrödinger operator in terms of eigenfunctions and applications

Cited by 5 publications

References 10 publications

A direct method for solving inverse Sturm–Liouville problems*

A direct method for solving inverse Sturm–Liouville problems*

A direct method for solving inverse Sturm-Liouville problems

A practical method for recovering Sturm-Liouville problems from the Weyl function

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