Semiclassical Analysis of Schrodinger Equations

本论文在国家科学研究资助计划的资助下，进行了以下四个方面的研究：1.微支撑在双曲线固定点上的传播（与J. 2.岛屿上的一口井产生的形状共振的假想部分（与A. L. Benbernou，A.Martinez合著）3.二维二能级Schr“odinger算子的锥交叉模型（与C.Lasser，4.一阶系统精确WKB方法的理论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。我们解决了这个问题，在分析和光滑的类别。Bony在巴黎的一次会议上讨论了这个结果，第二个结果是将Helffer-Sj“ostrand在解析势情况下的结果推广到smmoth势的情况，从岛的边界出现了焦散线。我们成功地将WKB解推广到焦散线之外，将其表示为Airy型积分形式，并将光滑相位和振幅近似解析地推广到复平面上.第三，得到了具有圆锥交叉本征势的二维二能级Schr“odinger算子共振态的量子化条件.我们把这个算子降为一维算子，并应用精确的WKB方法。我们已经写了一篇关于这个结果的论文。最后一个是对以前研究中所用方法的推广。将单Schr“odinger方程的精确WKB方法推广到二能级系统。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份文件。

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.