A direct method for solving inverse Sturm–Liouville problems*

Kravchenko, Vladislav V.; Torba, Sergii M.

doi:10.1088/1361-6420/abce9f

Cited by 18 publications

(38 citation statements)

References 28 publications

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“…The inverse problem consists in recovering q, h and H from given two spectra, {λ n } ∞ n=0 and {ν n } ∞ n=0 . Note that this problem coincides with the one considered in [24] if one changes x by π − x and h ↔ H. Now the equalities M (λ n ) = ∞ and M (ν n ) = 0, can be easily checked and give the reduction of the two spectra inverse problem to Problem 2.4.…”

Section: Two Spectramentioning

confidence: 58%

“…Having the Weyl function known completely, one can recover the sequence {λ n } ∞ n=0 as the sequence of its poles, and the sequence {α n } ∞ n=0 can be immediately obtained from the corresponding residuals, the formula (1.2.14) from [37] states that However the situation is more complicated if one looks for a practical method of recovering q, h and H from the Weyl function given only on a countable (even worse, finite) set of points from some (small) interval. The direct reduction to the inverse problems considered, in particular, in [24], is not possible. Let us formulate the corresponding problem and state some sufficient results for its unique solvability.…”

Section: Denote Additionallymentioning

confidence: 99%

“…In [20] (see also [21]) using representation (2.22) the Gelfand-Levitan equation was reduced to an infinite system of linear algebraic equations for the coefficients β 2n (x). In [24] a more accurate modification of the method was proposed. We present the final result from [24] below.…”

Section: Main System Of Equationsmentioning

confidence: 99%

“…In [24] a more accurate modification of the method was proposed. We present the final result from [24] below.…”

Section: Main System Of Equationsmentioning

confidence: 99%

“…That is, an efficient conversion of the original problem to an inverse problem by a given spectral density function is proposed. The method of solution introduced in [20], [21] and refined in [24] is adapted to recover the potential and the boundary conditions from the spectral density function.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Two Spectramentioning

confidence: 58%

Section: Denote Additionallymentioning

confidence: 99%

Section: Main System Of Equationsmentioning

confidence: 99%

“…In [24] a more accurate modification of the method was proposed. We present the final result from [24] below.…”

Section: Main System Of Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Kravchenko,

Torba

2021

Preprint

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Sinc method in spectrum completion and inverse Sturm–Liouville problems

Kravchenko,

Murcia‐Lozano

2024

Math Methods in App Sciences

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Cardinal series representations for solutions of the Sturm–Liouville equation with a complex‐valued potential are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to in any strip of the complex plane. This property of the obtained series representations leads to their applications in a variety of spectral problems. In particular, we show their applicability to the spectrum completion problem, consisting in computing large sets of the eigenvalues from a reduced finite set of known eigenvalues, without any information on the potential as well as on the constants from boundary conditions. Among other applications this leads to an efficient numerical method for computing a Weyl function from two finite sets of the eigenvalues. This possibility is explored in the present work and illustrated by numerical tests. Finally, based on the cardinal series representations obtained, we develop a method for the numerical solution of the inverse two‐spectra Sturm–Liouville problem and show its numerical efficiency.

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Spectrum completion and inverse Sturm–Liouville problems

Kravchenko

2022

Math Methods in App Sciences

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Given a finite set of eigenvalues of a regular Sturm–Liouville problem for the equation −y″+qfalse(xfalse)y=λy$$ -{y}^{{\prime\prime} }+q(x)y=\lambda y $$, the potential qfalse(xfalse)$$ q(x) $$ of which is unknown. We show the possibility to compute more eigenvalues without any additional information on the potential qfalse(xfalse)$$ q(x) $$. Moreover, considering the Sturm–Liouville problem with the boundary conditions y′false(0false)−hyfalse(0false)=0$$ {y}^{\prime }(0)- hy(0)=0 $$ and y′false(πfalse)+Hyfalse(πfalse)=0$$ {y}^{\prime}\left(\pi \right)+ Hy\left(\pi \right)=0 $$, where h,0.1emH$$ h,H $$ are some constants, we complete its spectrum without additional information neither on the potential qfalse(xfalse)$$ q(x) $$ nor on the constants h$$ h $$ and H$$ H $$. The eigenvalues are computed with a uniform absolute accuracy. Based on this result, we propose a new method for numerical solution of the inverse Sturm–Liouville problem of recovering the potential from two spectra. The method includes the completion of the spectra in the first step and reduction to a system of linear algebraic equations in the second. The potential qfalse(xfalse)$$ q(x) $$ is recovered from the first component of the solution vector. The approach is based on special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations possessing remarkable properties and leads to an efficient numerical algorithm for solving inverse Sturm–Liouville problems.

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A direct method for solving inverse Sturm–Liouville problems*

Cited by 18 publications

References 28 publications

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

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

Sinc method in spectrum completion and inverse Sturm–Liouville problems

Spectrum completion and inverse Sturm–Liouville problems

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