Spectral and Scattering Theory for Schrodinger Equations

薛定谔方程的谱和散射理论

• 批准号: 13640155
• 负责人: NAKAMURA Shu
• 金额: $ 2.43万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640155/
• 关键词: Schrodinger equation; scattering theory; semiclassical limit; random operators; tunneling effect; particle in magnetic field; smoothing effect; propagation of singularity; 関数解析; スペクトル理論; 波面集合; ランダムシュレディンガー作用素; スペクトルシフト関数

The aim of the project is to investigate differential equations of mathematical physics, in particular Schrodinger equations, using functional analysis and PDE methods. Here is a summary of results obtained, with emphasis on those obtained by the head investigator.(1)Semiclassical limit : The subject of semiclassical analysis is to study the behavior of the spectrum or the solutions to Schrodinger equation when the Planck constant tends to 0. The head investigator have been working on the tunneling effects in the phase space in collaboration with A. Martinez and V. Sordoni (Bologna Univ.). We apply our theory of phase space tunneling to the multi-state scattering in a joint paper of 2002, and also to the proof of an exponential estimate in the adiabatic limit in a joint paper iwth Sordoni. He also studied the relationship of resonances and scatteing in a joint work with Stefanov and Zworski.(2)Random Schrodinger operators : Schrodinger operator with potential that is a stochastic proce … More ss is called random Schrodinger operator, and it plays important roles in solid state physics. The head investigator have been working on the problem of the IDS (integrated density of states) and the Anderson localization for random Schrodinger operators. In a joint paper with Klopp, Nakano and Nomura, Schrodinger operator with random magnetic field is considered, and the localization of the spectrum is proved for a class of operators. A similar method was applied to so-called random hopping model to prove the localization in a joint work with Klopp (to appear). General methods to prove the uniqueness and continuity of the IDS are discussed in other papers of 2001 and 2002, partly in collaboration with Combes, Hislop and Klopp.(3)Propagation of singularity for Schrodinger euations : It is well-known that the propagation speed of solutions to the Schrodinger equation is infinite, and hence we cannot obtain propagation theorem as in the theory of wave equations. On the other hand, it is known that the decay of the initial state imply the smoothness of the solutions, and this is called smoothing effect. In a paper (to be published in Duke Math. J.), it is shown that the microlocal smoothing effect may be considered as propagation of (a sort of) wave front set, and the result is generalized to Schrodinger operator with long-range type perturbed principal symbol. Less

该项目的目的是使用泛函分析和偏微分方程方法研究数学物理的微分方程，特别是薛定谔方程。以下是所获得结果的摘要，重点是首席研究员获得的结果。(1)半经典极限：半经典分析的主题是研究当普朗克常数趋于0时光谱的行为或薛定谔方程的解。首席研究员一直与A. Martinez和V.合作研究相空间中的隧道效应。 Sordoni（博洛尼亚大学）。我们在 2002 年的联合论文中将我们的相空间隧道理论应用于多态散射，并在与 Sordoni 的联合论文中证明了绝热极限的指数估计。与 Stefanov 和 Zworski 合作研究了共振和散射的关系。(2)随机薛定谔算子：薛定谔算子具有潜在的随机过程…… More ss 被称为随机薛定谔算子，它在固体物理学中发挥着重要作用，首席研究员一直在与随机薛定谔算子的联合论文中研究 IDS（集成态密度）和安德森定位问题。 Klopp、Nakano 和 Nomura 考虑了具有随机磁场的薛定谔算子，并证明了一类算子的谱局域性。类似的方法被应用于所谓的随机跳变模型，以证明联合中的局域性。与 Klopp 合作（即将出现）。证明 IDS 的唯一性和连续性的一般方法在 2001 年和 2002 年的其他论文中讨论过，部分是与 Combes、Hislop 和 Klopp 合作的。(3)薛定谔方程的奇异性传播：众所周知，薛定谔方程解的传播速度是无限的，因此我们不能像波动方程理论那样得到传播定理。另一方面，已知初始状态的衰减意味着解的平滑性，这被称为平滑效应。在一篇论文（即将发表在 Duke Math. J.）中，表明微局部平滑效应可以。被认为是（一种）波前集的传播，并将结果推广到具有长程型扰动主符号的薛定谔算子。

项目成果

The functional connectome in obsessive-compulsive disorder: resting-state mega-analysis and machine learning classification for the ENIGMA-OCD consortium

• DOI: 10.1038/s41380-023-02077-0
• 发表时间: 2023-05-02
• 期刊: Molecular Psychiatry
• 影响因子: 11
• 作者: W. Bruin;Y. Abe;P. Alonso;A. Anticevic;Lea L. Backhausen;S. Balach;er;er;N. Bargalló;M. Batistuzzo;F. Benedetti;S. Bertolin Triquell;S. Brem;F. Calesella;B. Couto;D. Denys;M. A. N. Echevarria;G. Eng;S. Ferreira;Jamie D. Feusner;R. Grazioplene;P. Gruner;Joyce Y. Guo;Kristen Hagen;B. Hansen;Y. Hirano;M. Hoexter;N. Jahanshad;Fern Jaspers;S. Kasprzak;Minah Kim;K. Koch;Yoo Bin Kwak;J. Kwon;L. Lázaro;C. Li;C. Lochner;R. Marsh;I. Martínez;J. Menchón;P. Moreira;P. Morgado;A. Nakagawa;Tomohiro Nakao;J. Narayanaswamy;E. Nurmi;J. Zorrilla;J. Piacentini;M. Picó;F. Piras;F. Piras;C. Pittenger;J. Reddy;D. Rodriguez;Y. Sakai;Eiji Shimizu;V. Shivakumar;B. H. Simpson;C. Soriano;N. Sousa;G. Spalletta;E. Stern;S. Evelyn Stewart;P. Szeszko;Jinsong Tang;S. Thomopoulos;A. Thorsen;Y. Tokiko;Hirofumi Tomiyama;B. Vai;I. Veer;G. Venkatasubramanian;Nora C. Vetter;C. Vriend;S. Walitza;L. Waller;Zhen Wang;Anri Watanabe;N. Wolff;J. Yun;Qing Zhao;W. V. van Leeuwen;H. V. van Marle;L. van de Mortel;A. van der Straten;Y. D. van der Werf;P. Thompson;D. Stein;O. A. van den Heuvel;G. V. van Wingen
• 通讯作者: G. V. van Wingen