本文研究了$\R^n$，$n \geq 2$上径向薛定谔算子在固定能量下的逆散射问题。首先，我们考虑一类$\mathcal{A}$的潜力$q(r)$，它可以在$\Re z \geq 0$分析扩展，使$\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-ρ}$，$ρ\textgreater{} \frac{3}{2}$。如果$q$和$\tilde{q}$是两个这样的势，并且如果相应的相移$δ\_l$和$\tildeδ\_l$是超指数接近的，则$q=\tilde{q}$。其次，我们研究了一类可分解为$q(r)=q\_1(r) + q\_2(r)$的位势$q(r)$，使得$q\_1(r)$具有紧支集和$q\_2 (r) \in \mathcal{A}$。如果$q$和$\tilde{q}$是两个这样的潜力，我们表明，对于任何固定的$a\textgreater{}0$，${\ds{δ\_l - \tildeδ\_l \ = \ o \left( \frac{1}{l^{n-3}} \ \left( {\frac{ae}{2l}}\right)^{2l}\right)}}$当$l \rightarrow +\infty$当且仅当$q(r)=\tilde{q}(r)$几乎所有的$r \geq a$。这些证明在精神上与著名的Borg-Marchenko唯一性定理接近，并且严重依赖于Regge极点的局部化，Regge极点可以定义为复杂角动量平面中的共振。我们证明了对于非零的超指数下降势，Regge极点的个数总是无穷多，而且在右半平面上的任何垂直条中都不包含Regge极点.对于具有紧支集的势，我们能够明确地给出它们的渐近性。最后，对于在$\Re z \geq 0$中可以用$\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-ρ}$，$ρ\textgreater{}1$解析扩展的势，我们证明了Regge极点被限制在复平面上的一个垂直带中.
英文摘要：
We study inverse scattering problems at a fixed energy for radial Schrödinger 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 $\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-ρ}$, $ρ\textgreater{} \frac{3}{2}$. If $q$ and $\tilde{q}$ are two such potentials and if the corresponding phase shifts $δ\_l$ and $\tildeδ\_l$ 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 $a\textgreater{}0$,
${\ds{δ\_l - \tildeδ\_l \ = \ o \left( \frac{1}{l^{n-3}} \ \left( {\frac{ae}{2l}}\right)^{2l}\right)}}$ 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 $\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-ρ}$, $ρ\textgreater{}1$ , we show that the Regge poles are confined in a vertical strip in the complex plane.
On the ground state of the nonlinear Schr{ö}dinger equation: asymptotic behavior at the endpoint powers
CNRS IRMAR - UMR 6625（法国国家科学研究中心 IRMAR）；Univ. Lille, CNRS, Inria UMR 8524 - Laboratoire Paul Painlevé（里尔大学，法国国家科学研究中心，Inria UMR 8524 - 保罗·庞加莱实验室）；Department of Mathematics, McMaster University（麦克马斯特大学数学系）
On unique continuation for solutions of the Schr{ö}dinger equation on trees
我们证明了如果齐次树上的含时薛定谔方程的解在两个不同的时刻快速衰减，则该解是平凡的.对于自由薛定谔算子，我们利用Laplacian谱理论和复分析，得到了导致在任意时刻急剧衰减的初始条件的一个特征.然后，我们利用Colin de Verdi和Turc的紧支势的Schr dinger算子的谱分解，推广了我们在紧支势存在下的结果.最后，我们使用实变量的方法首先介绍了Escauriaza，Kenig，Ponce和Vega建立一个一般的尖锐的结果的情况下，有界的潜力。
