Resonances for steplike potentials: Forward and inverse results

Christiansen, T. J.

doi:10.1090/s0002-9947-05-03716-5

Cited by 32 publications

(17 citation statements)

References 21 publications

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“…The stability problem for one-dimensional Schrödinger operators was considered in the papers [20,28]. Moreover, there are other results about perturbations of the following model (unperturbed) potentials by compactly supported potentials: step potentials [3], periodic potentials [22], and linear potentials (corresponding to one-dimensional Stark operators) [25].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Inverse resonance scattering for Dirac operators on the half-line

Mokeev¹

2020

Preprint

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We consider massless Dirac operators on the half-line with compactly supported potentials. We solve the inverse problems in terms of Jost function and scattering matrix (including charactarization). We study resonances as zeros of Jost function and prove that a potential is uniquely determined by its resonances. Moreover, we prove the following:1) resonances are free parameters and a potential continuously depends on a resonance, 2) the forbidden domain for resonances is estimated, 3) asymptotics of resonance counting function is determined.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Inverse resonance scattering for Dirac operators on the half-line

Mokeev¹

2020

Preprint

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“…In these papers, the uniqueness, reconstruction, and characterization problems were solved, see also Zworski [42], Brown-Knowles-Weikard [4] concerning the uniqueness. Moreover, there are other results about perturbations of the following model (unperturbed) potentials by compactly supported potentials: step potentials [5], periodic potentials [24], and linear potentials (corresponding to one-dimensional Stark operators) [27].…”

Section: Introductionmentioning

confidence: 99%

Inverse resonance scattering on the real line

Korotyaev¹

2005

Inverse Problems

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We consider massless Dirac operators on the real line with compactly supported potentials. We solve two inverse problems (including characterization): in terms of zeros of reflection coefficient and in terms of poles of reflection coefficients (i.e. resonances). We prove that a potential is uniquely determined by zeros of reflection coefficients and there exist distinct potentials with the same resonances. We describe the set of "isoresonance potentials". Moreover, we prove the following: 1) a zero of the reflection coefficient can be arbitrarily shifted, such that we obtain the sequence of zeros of the reflection coefficient for an other compactly supported potential, 2) the forbidden domain for resonances is estimated, 3) asymptotics of resonances counting function is determined, 4) these results are applied to canonical systems. Contents 25 7. Canonical systems 29 References 36

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“…Inverse problems (characterization, recovering, uniqueness) for compactly supported potentials in terms of resonances were solved by Korotyaev for a Schrödinger operator with a compactly supported potential on the real line [15] and the half-line [13], see also Zworski [27], Brown-Knowles-Weikard [3] concerning the uniqueness. Moreover, there are other results about perturbations of the following model (unperturbed) potentials by compactly supported potentials: step potentials [5], periodic potentials [18], and linear potentials (corresponding to one-dimensional Stark operators) [19].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Discretization of inverse scattering on a half line

Korotyaev¹

2020

Preprint

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We solve inverse scattering problem for Schrödinger operators with compactly supported potentials on the half line. We discretize S-matrix: we take the value of the Smatrix on some infinite sequence of positive real numbers. Using this sequence obtained from S-matrix we recover uniquely the potential by a new explicit formula, without the Gelfand-Levitan-Marchenko equation.Deducated to Sergei Kuksin (Paris and Moscow) on the occasion of his 65-th birthday . Here n o (f ) is the multiplicity of 0 as a zero of a function f and note that n o (ψ) 1.

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Resonances for steplike potentials: Forward and inverse results

Abstract: Abstract. We consider resonances associated to the one dimensional Schrö-dinger operator −We obtain asymptotics of the resonance-counting function for several regions. Moreover, we show that in several situations, the resonances, V + , and V − determine V uniquely up to translation.

Cited by 32 publications

References 21 publications

Inverse resonance scattering for Dirac operators on the half-line

Inverse resonance scattering for Dirac operators on the half-line

Inverse resonance scattering on the real line

Discretization of inverse scattering on a half line

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