A Proof of the Local Borg-Marchenko Theorem

Bennewitz, Christer

doi:10.1007/s002200100384

Cited by 57 publications

(83 citation statements)

References 2 publications

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“…2 -Theorem 1.2 and the implication (A 1 ) =⇒ (A 2 ) in Theorem 1.4 follow in spirit the local Borg Marchenko's uniqueness theorem, (see [4,15,47,48]). Let us explain briefly our approach: we fix r > 0 and we define F (r, ν) as an application of the complex variable ν by: 19) where f ± (r, ν) andf ± (r, ν) are the Jost solutions associated with the potentials q andq.…”

Section: Q 2 (R) ∈ Amentioning

confidence: 70%

“…Hence, for ℜν > 0 large enough, we have δ(ν) =δ(ν). It follows from (2.12) that 4) or equivalently, using (2.11) and (2.12), α(ν)β(ν) −α(ν)β(ν) = 0 for ℜν > 0 large enough. By a standard analytic continuation, this last equality holds true for ℜν > 0.…”

Section: Uniqueness Of the Regge Interpolation Functionmentioning

confidence: 99%

“…It is well known that the Schrödinger equation (1.2) can be reduced to a countable family of radial equations, (see for instance [42]); indeed, we write: 4) and for functions u(x) = f (r)g(ω), where r =| x |> 0, ω = x r ∈ S n−1 , one has:…”

Section: Introductionmentioning

confidence: 99%

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Local Inverse Scattering at a Fixed Energy for Radial Schrödinger Operators and Localization of the Regge Poles

Daudé

Nicoleau

2015

Ann. Henri Poincaré

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We study inverse scattering problems at a fixed energy for radial Schrödinger operators on R n , n ≥ 2. First, we consider the class A of potentials q(r) which can be extended analytically in. If q andq are two such potentials and if the corresponding phase shifts δ l andδ l are super-exponentially close, then q =q. Secondly, we study the class of potentials q(r) which can be split into q(r) = q1(r) + q2(r) such that q1(r) has compact support and q2(r) ∈ A. If q andq are two such potentials, we show that for any fixed a > 0,when l → +∞ if and only if q(r) =q(r) for almost all r ≥ a. The proofs are close in spirit with the celebrated Borg-Marchenko uniqueness theorem, and rely heavily on the localization of the Regge poles that could be defined as the resonances in the complexified angular momentum plane. We show that for a non-zero super-exponentially decreasing potential, the number of Regge poles is always infinite and moreover, the Regge poles are not contained in any vertical strip in the right-half plane. For potentials with compact support, we are able to give explicitly their asymptotics. At last, for potentials which can be extended analytically in ℜz ≥ 0 with | q(z) |≤ C (1+ | z |) −ρ , ρ > 1 , we show that the Regge poles are confined in a vertical strip in the complex plane.

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Section: Q 2 (R) ∈ Amentioning

confidence: 70%

Section: Uniqueness Of the Regge Interpolation Functionmentioning

confidence: 99%

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Local Inverse Scattering at a Fixed Energy for Radial Schrödinger Operators and Localization of the Regge Poles

Daudé

Nicoleau

2015

Ann. Henri Poincaré

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“…It has also been used for inverse SturmLiouville theory (cf. [4,11,22]), again for standard boundary conditions. Prüfer introduced his angle in 1926 [19] as an alternative to Riccati equations for the study of Sturm-Liouville oscillation theory.…”

Section: Introductionmentioning

confidence: 99%

Equivalence of inverse Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter

Binding

Browne

Watson

2004

Journal of Mathematical Analysis and Applications

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Three inverse problems for a Sturm-Liouville boundary value problem −y + qy = λy, y(0) cos α = y (0) sin α and y (1) = f (λ)y(1) are considered for rational f . It is shown that the Weyl m-function uniquely determines α, f , and q, and is in turn uniquely determined by either two spectra from different values of α or by the Prüfer angle. For this it is necessary to produce direct results, of independent interest, on asymptotics and oscillation.

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“…In the following, we will adapt the principal ingredients of a recent proof of the local Borg-Marchenko uniqueness theorem for scalar Schrödinger operators (i.e., for m = 1), by Bennewitz [13], to the current Dirac-type situation. First we recall that by Lemma 5.2, (5.33) holds along the rays ρ +,j , j = 1, 2, for all α = (α 1 α 2 ) ∈ C m×2m satisfying (2.9).…”

Section: The Unique Weyl-titchmarsh Matrices Corresponding To the Halmentioning

confidence: 99%

Weyl–Titchmarsh $M$-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators

Clark

Gesztesy

2002

Trans. Amer. Math. Soc.

119

137

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Abstract. We explicitly determine the high-energy asymptotics for WeylTitchmarsh matrices associated with general Dirac-type operators on half-lines and on R. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operators and use it to prove a Borg-type theorem.

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A Proof of the Local Borg-Marchenko Theorem

Cited by 57 publications

References 2 publications

Local Inverse Scattering at a Fixed Energy for Radial Schrödinger Operators and Localization of the Regge Poles

Local Inverse Scattering at a Fixed Energy for Radial Schrödinger Operators and Localization of the Regge Poles

Equivalence of inverse Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter

Weyl–Titchmarsh $M$-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators

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