Spectral and Scattering Theory for Schrodinger Equations

该项目的目的是研究数学物理的微分方程，特别是薛定谔方程，使用泛函分析和偏微分方程组的方法。(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时薛定谔方程的谱或解的行为。首席研究员一直在与A.Martinez和V.Sordoni(博洛尼亚大学)合作研究相空间中的隧道效应。我们将相空间隧穿理论应用于2002年的一篇论文中的多态散射，并在与Sordoni的一篇论文中证明了绝热极限的指数估计。他还在与斯特凡诺夫和兹沃斯基的合作中研究了共振和散射的关系。(2)随机薛定谔算子：具有势的薛定谔算子是一个随机过程…更多的SS被称为随机薛定谔算符，它在固体物理中扮演着重要的角色。这位首席研究员一直致力于研究随机薛定谔算子的态密度问题和Anderson局部化问题。在Klopp，Nakano和Nomura的一篇联合论文中，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的局部性。一个类似的方法被应用于所谓的随机跳跃模型，在与Klopp(即将出现)的联合工作中证明了局部化。2001年和2002年的其他论文讨论了证明入侵检测系统唯一性和连续性的一般方法，部分是与Combes，Hislop和Klopp合作的。(3)薛定谔方程奇点的传播：众所周知，薛定谔方程解的传播速度是无穷的，因此我们不能像波动方程理论中那样得到传播定理。另一方面，已知初始状态的衰减意味着解的光滑性，这被称为光滑化效应。在一篇论文中(将在Duke Math上发表。J.)，证明了微局域光滑化效应可以看作是一种波前集的传播，并将结果推广到具有长程型扰动主元的薛定谔算子。较少

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