On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces

Савчук, А. М.; Шкаликов, А. А.

doi:10.1007/s11006-006-0204-6

Cited by 59 publications

(62 citation statements)

References 11 publications

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“…The key assertion is Theorem 1.3 (stated below in a convenient form) that these mappings are weakly nonlinear. (This theorem was proved for various problems in [46] and [47].) As we have mentioned already, the solution of Borg's problem for potentials q ∈ W α 2 in the entire scale α −1 was given by the authors in [45].…”

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confidence: 92%

Inverse problems for Sturm—Liouville operators with potentials in Sobolev spaces: Uniform stability

Савчук

Шкаликов

2010

Funct Anal Its Appl

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Two inverse problems for the Sturm-Liouville operator Ly = −y + q(x)y on the interval [0, π] are studied. For θ 0, there is a mapping F : W θ 2 → l θ B , F (σ) = {s k } ∞ 1 , related to the first of these problems, where W θ 2 = W θ 2 [0, π] is the Sobolev space, σ = q is a primitive of the potential q, and l θ B is a specially constructed finite-dimensional extension of the weighted space l θ 2 , where we place the regularized spectral data s = {s k } ∞ 1 in the problem of reconstruction from two spectra. The main result is uniform lower and upper bounds for σ − σ 1 θ via the l θ B -norm s − s 1 θ of the difference of regularized spectral data. A similar result is obtained for the second inverse problem, that is, the problem of reconstructing the potential from the spectral function of the operator L generated by the Dirichlet boundary conditions. The result is new even for the classical case q ∈ L 2 , which corresponds to θ = 1.Key words: Inverse Sturm-Liouville problem, singular potentials, stability for inverse problems.In the present paper we study two classical inverse problems for the Sturm-Liouville operatoron a finite interval. The first is the problem of reconstructing the potential from the two spectra of the operator (0.1) with Dirichlet and Dirichlet-Neumann boundary conditions, respectively. (We refer to it as Borg's problem.) The second is the problem of reconstructing the potential from the spectral function of the operator (0.1) with Dirichlet boundary conditions. (This operator will be called the Dirichlet operator.) The solution of these problems has long been known for the case of real potentials q ∈ L 2 ; in particular, a complete characterization of spectral data for potentials q of this class has been obtained. Our aim is to solve these problems for potentials q in the scale of Sobolev spaces W α 2 for any given α −1, including the case of α ∈ [−1, 0), where the potential is a singular function (a distribution). Here an important role is played by special Hilbert spaces that we construct to solve these problems. These spaces are needed to define and study mappings which we associate with these problems, as well as to completely describe (characterize) spectral data for potentials with primitive σ = q(t) dt ranging over the set of real functions in W α+1 2 .Once the inverse problems are solved, there arises an important question about a priori estimates, namely, the question of how small the change in a primitive of the potential q is in the norm of W α+1 2 when the spectral data undergo a change small in the norm of the corresponding Hilbert space, in which these data are placed. Earlier, a priori estimates have been obtained in the classical case (for α = 0). But these are estimates of local type, in which the constants, as well as the radius of the neighborhood where the estimates hold, depend on the potential q. The main goal of the present paper is to obtain uniform two-sided a priori estimates not only for the classical case α = 0 but also for all α > −1. The case α = −1 is exceptional...

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confidence: 92%

Inverse problems for Sturm—Liouville operators with potentials in Sobolev spaces: Uniform stability

Савчук

Шкаликов

2010

Funct Anal Its Appl

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“…228-229 and references] to explain numerical results. Savchuk and Shkalikov [28], strengthened these asymptotic results, obtaining expansions for eigenvalues of (1) with various boundary conditions, for all q in the Sobolev space W θ−1 2 , θ 0 . (When θ < 1 , q is a distribution.)…”

Section: Clues From Theoretical Resultsmentioning

confidence: 86%

Solving inverse Sturm-Liouville problems: theory and practice

Andrew

2017

ANZIAMJ

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Theoretical results on the solution of inverse Sturm-Liouville problems generally consider only idealized problems requiring much more data than is available in real applications. Typical theorems describe problems where infinitely many eigenvalues are known exactly, but in most applications we know only approximations of a finite, and usually small, number of eigenvalues. This paper considers how idealized theoretical results may assist practical numerical computation. It also reviews recent progress on a class of numerical methods for inverse Sturm-Liouville problems, it discusses some open questions, and it announces a new convergence result.

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“…Similar characterization of the set SD is available if p and r belong to W s 2 (0, 1) with s ≥ 0; cf. the results of [17,48] on eigenvalue asymptotics for Sturm-Liouville operators with potentials in Sobolev spaces. Secondly, the approach described is not restricted to the Dirichlet boundary conditions and can be used to reconstruct energy-dependent Sturm-Liouville equations under quite general separated boundary conditions.…”

Section: Reconstruction Of the Pencil: Uniquenessmentioning

confidence: 99%

Inverse spectral problems for energy-dependent Sturm–Liouville equations

Hryniv¹,

Pronska²

2012

Inverse Problems

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Abstract. We study the inverse spectral problem of reconstructing energy-dependent Sturm-Liouville equations from their Dirichlet spectra and sequences of the norming constants. For the class of problems under consideration, we give a complete description of the corresponding spectral data, suggest a reconstruction algorithm, and establish uniqueness of reconstruction. The approach is based on connection between spectral problems for energy-dependent Sturm-Liouville equations and for Dirac operators of special form.

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On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces

Cited by 59 publications

References 11 publications

Inverse problems for Sturm—Liouville operators with potentials in Sobolev spaces: Uniform stability

Inverse problems for Sturm—Liouville operators with potentials in Sobolev spaces: Uniform stability

Solving inverse Sturm-Liouville problems: theory and practice

Inverse spectral problems for energy-dependent Sturm–Liouville equations

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