Semiclassical Analysis of Schroedinger equations

这一时期的研究主要集中在以下几个方面。首先，与j.f。我考虑了具有双曲不动点和相应的进出稳定流形的哈密顿量。在一般假设下，我们证明了输出稳定流形（输出数据）上对应的薛定谔方程的微局部解是唯一地由输入稳定流形（输入数据）上的微局部解决定的。此外，我们成功地将输出数据用输入数据描述为傅里叶积分算子，其相位和振幅由几何量显式给出。这些结果写在预印本“伪微分算子在双曲不动点的微局部核”中。其次，在与a . L. Benbernou和a . Martinez的合作中，我考虑了由“岛上的井”产生的形状共振宽度的渐近扩展。大约20年前，Helffer和Sjostrand展示了解析势的经典扩展，其前因子呈指数级小，其速率由从井到海的Agmon距离给出。我们只对光滑势进行了同样的推测。

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.