Semiclassical Analysis of Schrodinger Equations

薛定谔方程的半经典分析

• 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
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540149/
• 关键词: Schr"odinger Equation; WKB method; Semicalssical analysis; Resonance; Monodromy operator; Schrodinger方程式; モノドロミー作要素; ホモクリニック軌道; 超局所解析

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.

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. classer，L.NENDELEC）4。一阶系统的精确WKB方法（使用L.NENDELEC）。第一个问题是从稳定的歧管到与双曲线固定点相关的不稳定歧管的微支持者的传播。对于Smmoth势的分析潜力，Helffer-SJ“ Ostrand。我们成功地将WKB解决方案扩展到了苛性碱中，以通风型的突出形式将其表示，并通过几乎分析扩展到复杂平面扩展光滑的相位和振幅。第三个是获得2-Dim 2级Schr的共振的量化条件。写了一篇有关此结果的论文。最后一个是对先前研究中使用的方法的概括。它概括了单个Schr“ Odinger”方程式的确切WKB方法的理论到2级系统。我谈到了在京都举行的国际大会上谈论的结果，我们现在正在准备一篇论文。

项目成果

S.Fujiie, M.Zerzeri: "Bohr-Sommerfeld quantization condition derived by a microlocal WKB method"Proceedings of ICONA-MECOM 2003 Vietnam Journal of Mathematics. (to appear).

S.Fujiie, M.Zerzeri：“Bohr-Sommerfeld 量子化条件由微局域 WKB 方法导出”ICONA-MECOM 2003 越南数学杂志论文集。

S.Fujiie, T.Ramond: "Breit-Wigner formulas at barrier tops"Journal of Mathematical Physics. 44-5. 1971-1983 (2003)

S.Fujiie、T.Ramond：“势垒顶部的 Breit-Wigner 公式”数学物理杂志。

書評:Dimassi-Sjostrand, "Spectral Asymptotics in the Semiclassical Analysis"

书评：Dimassi-Sjostrand，“半经典分析中的谱渐近论”

• 发表时间: 2005
• 期刊: 数学(日本数学会) 57-1
• 作者: 藤家 雪朗

Book-Review : Dimassi-Sjostrand,"Spectral Asymptotics in the Semiclassical Analysis"H.Chihara

书评：Dimassi-Sjostrand，“半经典分析中的谱渐近论”H.Chihara

• 发表时间: 2005
• 期刊: Sugaku, Japanese Mathematical Society 57-1
• 作者: 藤家 雪朗;千原 浩之;S.Fujii'e
• 通讯作者: S.Fujii'e

Third order semilinear dispersive equations related to deep water

与深水有关的三阶半线性色散方程

• 期刊: Trans.Amer.Math.Soc. (In press)
• 作者: H.Mizuno;I.Sato;I.Sato;千原 浩之
• 通讯作者: 千原 浩之