Francois Nicoleau (Nantes)

Local inverse scattering at a fixed energy for radial Schrodinger operators and localization of the Regge poles. 

Abstract: We study inverse scattering problems at a fixed energy for radial Schrodinger 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 $\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}$, $\rho > \frac{3}{2}$. 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 $a>0$, ${\ds{\delta_l - \tilde{\delta}_l \ = \ o \left( \frac{1}{l^{n-3}} \ \left( {\frac{ae}{2l}}\right)^{2l}\right)}}$ 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. This is a joint work with Thierry Daude (Universite de Cergy-Pontoise).