Inverse Spectral Theory for Some Singular Sturm-Liouville Problems

Carlson, Robert

doi:10.1006/jdeq.1993.1102

Cited by 54 publications

(61 citation statements)

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“…The behaviour of solutions of (1.a) near x = 0 is described in [6]. Lemmas 2.1 and 2.3 of that paper show that (1.a) has a unique solution, which will be denoted y 2 (x, λ, p), satisfying…”

Section: Lemma 22mentioning

confidence: 99%

“…The normalization is chosen for later convenience. The subscript 2 is used to maintain notation consistent with previous work [6,13,30]. By using variation of parameters, any solution of (2.a) may be represented as a solution of the integral equation…”

Section: Solutions Of the Unperturbed Equation (1c)mentioning

confidence: 99%

“…The division by x m+1 still leaves us with functions continuous on [0, 1] by Lemma 2.1 of [6]. Letting…”

Section: Lemma 22mentioning

confidence: 99%

“…This theory was extended to the case m = 1 in [13], and to nonnegative integers m in [6,7]. The goal of this paper is to extend certain of the direct and inverse spectral theoretic results to the cases m ≥ −1/2.…”

Section: Introductionmentioning

confidence: 99%

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A Borg–Levinson theorem for Bessel operators

Carlson¹

1997

Pacific J. Math.

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This paper presents a direct analog of the Borg-Levinson theorem on the recovery of a potential from the sequence of eigenvalues and norming constants for differential equations of the formon the unit interval subject to various boundary conditions. This result is used to show that even zonal Schrödinger operators and Laplace operators on spheres are uniquely determined by a subsequence of their eigenvalues.

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Section: Lemma 22mentioning

confidence: 99%

Section: Solutions Of the Unperturbed Equation (1c)mentioning

confidence: 99%

“…The division by x m+1 still leaves us with functions continuous on [0, 1] by Lemma 2.1 of [6]. Letting…”

Section: Lemma 22mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Borg–Levinson theorem for Bessel operators

Carlson¹

1997

Pacific J. Math.

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“…For ≥ −1/2, there are much more generalized results from [28,29]. In particular, the solution of (15) is an entire function of order one and of finite type [4,9,10,14,[27][28][29][30] that has a Sturm-Liouville type of spectral analysis. Now we fulfill the boundary conditions in (1).…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

On Certain Translation Invariant Properties of Interior Transmission Spectra and Their Doppler’s Effect

Chen

2017

Advances in Mathematical Physics

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We study the translation invariant properties of the eigenvalues of scattering transmission problem. We examine the functional derivative of the eigenvalue density function Δ(̂) to the defining index of refraction ( ). By the limit behaviors in frequency sphere, we prove some results on the inverse uniqueness of index of refraction. In physics, Doppler's effect connects the variation of the frequency/eigenvalue and the motion velocity/variation of position variable. In this paper, we proved the functional derivative Δ(̂) = (1 + √ ( (̂)))/ .

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Commutation methods for Schrödinger operators with strongly singular potentials

Kostenko

Sakhnovich

2011

Mathematische Nachrichten

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Abstract. We explore the connections between singular Weyl-Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in terms of the original operator. We apply the results to spherical Schrödinger operators (also known as Bessel operators). We also investigate the connections with the generalized Bäcklund-Darboux transformation.

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Inverse Spectral Theory for Some Singular Sturm-Liouville Problems

Cited by 54 publications

References 0 publications

A Borg–Levinson theorem for Bessel operators

A Borg–Levinson theorem for Bessel operators

On Certain Translation Invariant Properties of Interior Transmission Spectra and Their Doppler’s Effect

Commutation methods for Schrödinger operators with strongly singular potentials

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