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

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.