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

TL;DR

This paper investigates inverse scattering at fixed energy for radial Schrödinger operators, establishing uniqueness results and analyzing the distribution of Regge poles, with implications for potential reconstruction and resonance behavior.

Contribution

It introduces new uniqueness theorems for inverse scattering with super-exponentially decreasing potentials and characterizes the asymptotic distribution of Regge poles.

Findings

Super-exponential closeness of phase shifts implies potential equality.

Regge poles are infinite for non-zero super-exponentially decreasing potentials.

Regge poles are confined in a vertical strip for certain analytic potentials.

Abstract

We study inverse scattering problems at a fixed energy for radial Schr\"{o}dinger operators on $\R^n$, $n \geq 2$. First, we consider the class $\mathcal{A}$ of potentials $q(r)$ which can be extended analytically in $\Re z \geq 0$ such that $\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}$, $\rho \textgreater{} \frac{3}{2}$. 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 $a\textgreater{}0$, ${\ds{\delta\_l - \tilde{\delta}\_l \ = \ o \left( \frac{1}{l^{n-3}} \ \left( {\frac{ae}{2l}}\right)^{2l}\right)}}$ when $l \rightarrow +\infty$ if and only if $q(r)=\tilde{q}(r)$ for almost all $r \geq a$. 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 $\Re z \geq 0$ with $\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}$, $\rho \textgreater{}1$ , we show that the Regge poles are confined in a vertical strip in the complex plane.