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Semiclassical Analysis of Schroedinger equations

薛定谔方程的半经典分析

基本信息

批准号：
17540141
负责人：
FUJIIE Setsuro
金额：
$ 2.11万
依托单位：
University of Hyogo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540141/
关键词：
semi-classical analysis WKB method microlocal analysis resonance Schroedinger equation hyperbolic fixed point almost analytic extension propagation of siugularity 錐状交差ポテンシャル

项目摘要

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.

这一时期的主要研究成果如下：第一，与J. F.博尼，T。Ramond和M. Zerzeri，我考虑了一个双曲不动点和相应的传入和传出稳定流形的哈密顿量。我们表明，在一个通用的假设下，相应的薛定谔方程的微局部解的输出稳定的流形（输出数据）是唯一确定的传入的稳定的流形（输入数据）。此外，我们成功地描述了输出数据的输入数据的傅里叶积分算子，其相位和振幅的几何量显式地给出。这些结果被写在预印本“双曲不动点上伪微分算子的微局部核”中。L. Benbernou和A.马丁内斯，我认为渐近扩展的形状共振的宽度所产生的“一个岛”。大约20年前，Helffer和Sjostrand证明了解析势的经典展开，其前因子是指数小的，其速率由从井到海的Agmon距离给出。我们推测仅对于光滑势也有相同的结果。

项目成果

期刊论文数量（13）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Third order semilinear dispersive equations related to deep water waves

与深水波有关的三阶半线性色散方程

DOI：
发表时间：
期刊：
Trans. Amer. Math. Soc. (To appear)
影响因子：
0
作者：
Ohnishi;M.;Tsujimura;M.;Arisawa Mariko;S.Fujiie;S.Fujiie;S.Fujiie;S.Fujiie;S.Fujiie;S.Doi;H.Chihara;S.Fujiie;H.Chihara;S.Doi;H.Chihara
通讯作者：
H.Chihara

An exact WKB method for 2×2 systems and applications

适用于 2×2 系统和应用的精确 WKB 方法