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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.

感谢科学研究补助金，我做了以下 4 项研究：1.双曲固定点处微支撑的传播（与 J.-F.Bony、T.Ramond、M.Zerzeri 合作）2.由岛中的井产生的形状共振的虚部（与 A.L.Benbernou、A.Martinez 合作）3.2 维 2 水平的圆锥形交叉模型薛定谔算子（有 C.Lasser，L.Nedelec）4.一阶系统的精确WKB方法理论（与L.Nedelec）。第一个问题是关于微支撑从稳定流形到与双曲不动点相关的不稳定流形的传播。我们在解析类别和平滑类别中解决了这个问题。博尼在巴黎的一次大会上谈到了这个结果。第二个是结果的延伸 Helffer-Sj"ostrand在解析势的情况下到平滑势的情况下。从岛的边界出现焦散。我们成功地将 WKB 解扩展到焦散之外，以艾里型积分的形式表示，并通过几乎解析扩展到复平面来扩展平滑相位和幅度。第三个是获得具有圆锥交叉特征势的 2 维 2 级 Schr"odinger 算子的谐振量化条件。我们将该算子简化为 1 维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。最后是对前人研究3中所用方法的推广。它将单薛定谔方程的精确WKB方法的理论推广到2级系统。我在京都举行的国际会议上谈到了这个结果，我们现在正在准备一篇论文。