Semiclassical Analysis of Schroedinger equations

首先，我与J.-F.Bony，T.Ramond和M.Zerzeri合作，研究了一类具有双曲不动点的哈密顿算子以及相应的传入和传出稳定流形。在一般假设下，我们证明了对应的薛定谔方程在流出稳定流形(输出数据)上的微局部解是由传入稳定流形(输入数据)上的微局部解唯一确定的。此外，我们成功地将输出数据描述为傅里叶积分算子，其相位和幅度由几何量明确地给出。这些结果被写在预印本《双曲不动点伪微分算子的微局部核》一书中。第二，我与A.L.Benbernou和A.Martinez合作，考虑了由一座岛上的井产生的形状共振宽度的渐近展开。大约20年前，Helffer和Sjostrand对于解析势证明了它具有一个具有指数小的前因子的经典展开式，其速率由从井到海的Agmon距离给出。我们猜测，只有光滑的势才有同样的结果。

The main researches during this period are the fallowings.First, in collaboration with J.-F. Bony, T. Ramond and M. Zerzeri, I considered a Hamiltonian with a hyperbolic fixed point and the corresponding incoming and outgoing stable manifolds. We showed, under a generic assumption, that the microlocal solution of the corresponding Schroedinger equation on the outgoing stable manifold (output data) is uniquely determined by that on the incoming stable manifold (input data). Moreover, we succeeded in describing the output data in terms of the input data as Fourier integral operator, whose phase and amplitude are explicitely given by geometrical quantities. These results are written in the preprint "Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point.Second, in collaboration with A. L. Benbernou and A. Martinez, I considered the asymptotic expansion of the width of shape resonances created by a "well in an island". About 20 years ago, Helffer and Sjostrand showed for analytic potentials that it has a classical expansion with an exponentially small prefactor whose rate is given by the Agmon distance from the well to the sea. We conjectured the same result for only smooth potentials.