Local inverse scattering at a fixed energy for radial Schr{ö}dinger operators and localization of the Regge poles

Abstract

We study inverse scattering problems at a fixed energy for radial Schrödinger operators on ${\mathbb{R}}^n$, $n\ge 2$. First, we consider the class $\mathcal{A}$ of potentials $q(r)$ which can be extended analytically in $\Re z\ge 0$ such that $\mid q(z)\mid \le C(1+\mid z\mid {)}^{-\rho }$, $ρ\textgreater{} \frac{3}{2}$. If $q$ and $\tilde{q}$ are two such potentials and if the corresponding phase shifts $\delta \_l$ and $\tilde{\delta }\_l$ are super-exponentially close, then $q=\tilde{q}$. Secondly, we study the class of potentials $q(r)$ which can be split into $q(r)=q\_1(r)+q\_2(r)$ such that $q\_1(r)$ has compact support and $q\_2(r)\in \mathcal{A}$. If $q$ and $\tilde{q}$ are two such potentials, we show that for any fixed $a\textgreater{}0$, ${\ds{δ\_l - \tildeδ\_l \ = \ o \left( \frac{1}{l^{n-3}} \ \left( {\frac{ae}{2l}}\right)^{2l}\right)}}$ when $l\to +\infty$ if and only if $q(r)=\tilde{q}(r)$ for almost all $r\ge a$. 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 $\Re z\ge 0$ with $\mid q(z)\mid \le C(1+\mid z\mid {)}^{-\rho }$, $ρ\textgreater{}1$ , we show that the Regge poles are confined in a vertical strip in the complex plane.

52 pages

References in corpus (4)

• On Local Borg-Marchenko Uniqueness Results
• Transparent Potentials at Fixed Energy in Dimension Two. Fixed-Energy Dispersion Relations for the Fast Decaying Potentials
• Inverse scattering problem with part of the fixed-energy phase shifts
• On Inverse Scattering at a Fixed Energy for Potentials with a Regular Behaviour at Infinity

Cited by in corpus (2)

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