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Article dans une revue Annales Henri Poincaré Année : 2016

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 $\R^n$, $n \geq 2$. 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 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 $\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}$, $\rho >1$ , we show that the Regge poles are confined in a vertical strip in the complex plane.
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Dates et versions

hal-01114204, version 1 (08-02-2015)
hal-01114204, version 2 (12-06-2015)

Identifiants

  • HAL Id : hal-01114204 , version 2
  • ARXIV : 1502.02276

Citer

Thierry Daudé, Francois Nicoleau. Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles. Annales Henri Poincaré, 2016, 17 (10), pp. 2849-2904. ⟨hal-01114204v2⟩
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