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Semiclassical Analysis of Schrodinger Equations

薛定谔方程的半经典分析

基本信息

批准号：
15540149
负责人：
FUJIIE Setsuro
金额：
$ 2.24万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

项目摘要

Thanks to the Grant-In-Aid for Scientific Research, I did the following 4 researches :1.Propagation of the microsupport at a hyperbolic fixed point (with J.-F.Bony, T.Ramond, M.Zerzeri)2.Imaginary part of shape resonances created by a well in an island (with A.L.Benbernou, A.Martinez)3.A conically crossing model for 2-dim 2-level Schr"odinger operators (with C.Lasser, L.Nedelec)4.Theory of exact WKB method for first order systems (with L.Nedelec).The first problem is about the propagation of the microsupport from the stable manifold to the unstable manifold associated with the hyperbolic fixed point. We solved this problem in both analytic and smooth categories. Bony has talked about this result in a-congress in Paris.The second is an extension of the result by Helffer-Sj"ostrand in the case of analyhtic potential to the case of smmoth potential.There appears a caustics from the boundary of the island. We succeeded to extend a WKB solution beyond the caustics by representing it in the form of Airy type intagral and extending the smooth phase and the amplitude by almost analytic extension to the complex plane.The third is to obtain the quantization condition of resonances of the 2-dim 2-level Schr"odinger operator with conically crossing eigenpotentials. We reduced this operator to a 1-dim one and applied the exact WKB method. We have already written a paper about this result.The last is a generalization of the method used in the previous research 3. It generalizes the theory of exact WKB method for single Schr"odinger equations to 2-level systems. I talked about this result in an international congress held in Kyoto and we are now preparing a paper.

得益于科研资助，我做了以下4项研究：1。微支撑在双曲不动点的传播(与j - f。波尼，t .雷蒙德，m .泽泽里孤岛上的井所产生的形状共振的虚部（与A.L.Benbernou, a.m inez）2-dim 2-level Schr ' odinger算子的二次交叉模型（with C.Lasser, l.n nedelec）一阶系统的精确WKB方法理论（与L.Nedelec合著）。第一个问题是与双曲不动点相关的微支撑从稳定流形到不稳定流形的传播问题。我们在解析类和光滑类中解决了这个问题。博尼在巴黎的一次大会上谈到了这一结果。第二种是将Helffer-Sj"ostrand在解析势情况下的结果推广到光滑势情况。从岛的边界出现了一个焦散。我们成功地将WKB解用Airy型积分的形式表示，并将光滑相位和振幅通过几乎解析扩展到复平面，从而将其扩展到焦散之外。第三是得到具有圆锥交叉特征势的2-dim 2-能级Schr ' odinger算子共振的量子化条件。我们将这个算子简化为1-dim算子，并应用了精确的WKB方法。我们已经写了一篇关于这个结果的论文。最后是对前面研究方法的推广。将单Schr"odinger方程的精确WKB方法理论推广到二能级系统。我在京都举行的一次国际会议上谈到了这一结果，我们现在正在准备一篇论文。