A uniqueness result for one‐dimensional inverse scattering

Bennewitz, Christer; Brown, B. M.; Weikard, Rudi

doi:10.1002/mana.201100101

Cited by 3 publications

(7 citation statements)

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“…In addition, assume that R is a real-valued function of bounded variation on [0, ω] and define r as in (4. 19), (4.20). Then K * rK has infinitely many positive (resp., negative) eigenvalues unless R + (resp., R − ) is a pure jump function with only finitely many jumps (if any).…”

Section: The Case Of Periodic Coefficientsmentioning

confidence: 99%

“…In the specific context of the left-definite Camassa-Holm spectral problem we refer to [9], [10], [17], [19], [30], [31], [32], [33], [34] [35], [37], [51], [62], [85], [94], [96], [106], [107], [108], [109], and the literature cited therein.…”

Section: Introductionmentioning

confidence: 99%

“…There exists an enormous body of literature for these kinds of generalized spectral problems and without any possibility of achieving completeness, we refer, for instance, to [2], [12], [55], [68], [72], [73], [74], [75], [98], [116], [131], and the extensive literature cited therein in the context of general boundary value problems. In the context of indefinite Sturm-Liouville-type boundary value problems we mention, for instance, [6], [8], [11], [13], [14], [15], [16], [18], [19], [20], [21], [23], [30], [31], [32], [37], [47], [52], [56], [83], [84], [86], [89], [90], [91], [99], [100], [117], [132], [135,Chs. 5,11,12], and again no attempt at a comprehensive account of the existing literature is possible due to the enormous volume of the latter.…”

Section: Introductionmentioning

confidence: 99%

“…Eliminating φ 2 in (1.4) then results in the scalar (weighted) spectral problem − φ ′′ (z, x) + φ(z, x) = z[u xx (x) − 4u(x)]φ(z, x), (z, x) ∈ (C\{0}) × R. (1.6) In the specific context of the left-definite Camassa-Holm spectral problem we refer to [9], [10], [17], [19], [30], [31], [32], [33], [34] [35], [37], [51], [62], [85], [94], [96], [106], [107], [108], [109], and the literature cited therein.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, in the context of the CH spectral problem (1.6) this has briefly been used, for instance, in [36] (in the periodic case), in [33] (in the context of the CH scattering problem), in [62] (in connection with real-valued algebro-geometric CH solutions), and in [107] (in connection with CH flows and Fredholm determinants). However, apart from the approach discussed in [18], [19], [20], most investigations associated with the CH spectral problem (1.6) appear to focus primarily on certain Liouville-Green transformations which transform (1.6) into a Schrödinger equation for some effective potential coefficient (see, e.g., [30], [31], [32]). This requires additional assumptions on the coefficients which in general can be avoided in the context of (1.18).…”

Section: Introductionmentioning

confidence: 99%

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Some Remarks on the Spectral Problem Underlying the Camassa–Holm Hierarchy

Gesztesy

Weikard

2014

Operator Theory: Advances and Applications

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Dedicated with great pleasure to Ludwig Streit on the occasion of his 75th birthday.Abstract. We study particular cases of left-definite eigenvalue problems Aψ = λBψ, with A ≥ εI for some ε > 0 and B self-adjoint, but B not necessarily positive or negative definite, applicable, in particular, to the eigenvalue problem underlying the Camassa-Holm hierarchy. In fact, we will treat a more general version where A represents a positive definite Schrödinger or Sturm-Liouville operator T in L 2 (R; dx) associated with a differential expression of the formrepresents an operator of multiplication by r(x) in L 2 (R; dx), which, in general, is not a weight, that is, it is not nonnegative (or nonpositive) a.e. on R. In fact, our methods naturally permit us to treat certain classes of distributions (resp., measures) for the coefficients q and r and hence considerably extend the scope of this (generalized) eigenvalue problem, without having to change the underlying Hilbert space L 2 (R; dx). Our approach relies on rewriting the eigenvalue problem Aψ = λBψ in the formand a careful study of (appropriate realizations of) the operator A −1/2 BA −1/2 in L 2 (R; dx).In the course of our treatment, we review and employ various necessary and sufficient conditions for q to be relatively bounded (resp., compact) and relatively form bounded (resp., form compact) with respect to T 0 = −d 2 /dx 2 defined on H 2 (R). In addition, we employ a supersymmetric formalism which permits us to factor the second-order operator T into a product of two firstorder operators familiar from (and inspired by) Miura's transformation linking the KdV and mKdV hierarchy of nonlinear evolution equations. We also treat the case of periodic coefficients q and r, where q may be a distribution and r generates a measure and hence no smoothness is assumed for q and r.

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Section: The Case Of Periodic Coefficientsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some Remarks on the Spectral Problem Underlying the Camassa–Holm Hierarchy

Gesztesy

Weikard

2014

Operator Theory: Advances and Applications

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Compactness of iso-resonant potentials for Schrödinger operators in dimensions one and three

Hislop¹,

Wolf²

2021

Trans. Amer. Math. Soc.

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We prove compactness of a restricted set of real-valued, compactly supported potentials V V for which the corresponding Schrödinger operators H V H_V have the same resonances, including multiplicities. More specifically, let B R ( 0 ) B_R(0) be the ball of radius R > 0 R > 0 about the origin in R d \mathbb {R}^d , for d = 1 , 3 d=1,3 . Let I R ( V 0 ) \mathcal {I}_R (V_0) be the set of real-valued potentials in C 0 ∞ ( B ¯ R ( 0 ) ; R ) C_0^\infty ( \overline {B}_R(0); \mathbb {R}) so that the corresponding Schrödinger operators have the same resonances, including multiplicities, as H V 0 H_{V_0} , for a fixed, but arbitrary, potential V 0 ∈ C 0 ∞ ( B ¯ R ( 0 ) ; R ) V_0 \in C_0^\infty ( \overline {B}_R(0); \mathbb {R}) . We prove that the set I R ( V 0 ) \mathcal {I}_R (V_0) is a compact subset of C 0 ∞ ( B ¯ R ( 0 ) ) C_0^\infty (\overline {B}_R(0)) in the C ∞ C^\infty -topology. An extension to Sobolev spaces of less regular potentials is discussed.

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Compactness of iso-resonant potentials for Schrödinger operators in dimensions one and three

Hislop¹,

Wolf²

2018

Preprint

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We prove compactness of a restricted set of real-valued, compactly supported potentials V for which the corresponding Schrödinger operators HV have the same resonances, including multiplicities. More specifically, let BR(0) be the ball of radius R > 0 about the origin in R d , for d = 1, 3. Let IR(V0) be the set of real-valued potentials in C ∞ 0 (BR(0); R) so that the corresponding Schrödinger operators have the same resonances, including multiplicities, as HV 0 . We prove that the set IR(V0) is a compact subset of C ∞ 0 (BR(0)) in the C ∞ -topology. An extension to Sobolev spaces of less regular potentials is discussed.

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A uniqueness result for one‐dimensional inverse scattering

Cited by 3 publications

References 15 publications

Some Remarks on the Spectral Problem Underlying the Camassa–Holm Hierarchy

Some Remarks on the Spectral Problem Underlying the Camassa–Holm Hierarchy

Compactness of iso-resonant potentials for Schrödinger operators in dimensions one and three

Compactness of iso-resonant potentials for Schrödinger operators in dimensions one and three

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