Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles
Résumé
We study inverse scattering problems at a fixed energy for radial Schr\"{o}dinger operators on $\mathbb{R}^n$, $n \geq 2$. First, we consider the class of potentials $q(r)$ which can be extended analytically in $\Re z \geq 0$ such that $\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}$, $\rho > \frac{3}{2}$. 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 $q_2 (r)$ can be extended analytically in $Re \ z \geq 0$ with $\mid q_2(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}$, $\rho > 2$.
If $q$ and $\tilde{q}$ are two such potentials, we show that for any fixed $a>0$,
${\displaystyle{\delta_l - \tilde{\delta}_l \ = \ O \left( \frac{1}{l^{n-1}} \ \left( {\frac{ae}{2l}}\right)^{2l}\right)}}$ if and only if $q(r)=\tilde{q}(r)$ for 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 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 $q(z) =O((1+\mid z\mid)^{-\rho})$ with $\rho >1$, we show that the Regge poles are confined in a vertical strip in the complex plane.
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