Eigenvalue asymptotics for Sturm–Liouville operators with singular potentials

Hryniv, Rostyslav; Mykytyuk, Yaroslav V.

doi:10.1016/j.jfa.2006.04.015

Cited by 36 publications

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References 26 publications

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“…Hryniv and Mykytyuk ( [18], [21]) proved the existence of a transformation operator for equations with such potentials and gave solutions of the classical inverse problems for potentials q ∈ W −1 2 (see, e.g., [19], [20], and [22]). Marchenko and Ostrovskii ([36], [34]) described the spectral data of Borg's problem for potentials q in the Sobolev spaces W α 2 with integer smoothness indices α = 0, 1, 2, .…”

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

Савчук

Шкаликов

2010

Funct Anal Its Appl

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“…For the sake of definiteness, we shall concentrate on the case h 0 = h 1 = ∞ and h 0 = h ∈ R only, the other cases being analogous. More precisely, we shall formulate necessary and sufficient conditions on sequences (λ n ) and (µ n ) in order that they should be eigenvalues of the Sturm-Liouville operators T (q, ∞, ∞) and T (q, h, ∞), respectively, for some choice of q in W s−1 2 (0, 1) and h ∈ R. For an arbitrary intermediate value s ∈ (0, 1), the direct spectral problem was studied in [10,13,26]. For instance, it was proved in [10] that the eigenvalue remaindersλ n andμ n defined in (1.5) are, respectively, even and odd sine Fourier coefficients of some function from W s 2 (0, 1) (cf.…”

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confidence: 99%

“…the above-mentioned cases s = 0 in (A2 ) and s = 1 in (A2)). More exactly, the main result from [10] reads as follows. As an intermediate step we solve the inverse spectral problem of recovering the potential of a Sturm-Liouville expression from its Dirichlet spectrum (λ n ) and the so-called norming constants (α n ).…”

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Inverse Spectral Problems for Sturm–liouville Operators With Singular Potentials. Iv. Potentials in the Sobolev Space Scale

Hryniv¹,

Mykytyuk²

2006

Proceedings of the Edinburgh Mathematical Society

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“…and developed by A. Savchuk and A. Shkalikov [25,27] (see also [26,28,29]) and R. Hryniv and Ya. Mykytyuk [8] (see also [9]- [13]). For specific potentials see W. N. Everitt and A. Zettl [6,7].…”

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Fourier Method for One-dimensional Schrödinger Operators with Singular Periodic Potentials

Djakov¹,

Mityagin²

2010

Topics in Operator Theory

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Eigenvalue asymptotics for Sturm–Liouville operators with singular potentials

Cited by 36 publications

References 26 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

Inverse Spectral Problems for Sturm–liouville Operators With Singular Potentials. Iv. Potentials in the Sobolev Space Scale

Fourier Method for One-dimensional Schrödinger Operators with Singular Periodic Potentials

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