Spectral and Scattering Theory for Schrodinger Equations

该项目的目的是利用泛函分析和偏微分方程方法研究数学物理中的微分方程，特别是薛定谔方程。以下是所获得结果的摘要，重点是由首席调查员获得的结果。(1)半经典极限：半经典分析的主题是研究普朗克常数趋于0时谱或薛定谔方程解的行为。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。在2002年的一篇联合论文中，我们将相空间隧穿理论应用于多态散射，并在与Sordoni合著的一篇论文中应用于绝热极限下的指数估计的证明。他还与Stefanov和Zworski合作研究了共振和散射的关系。(2)随机薛定谔算子：具有随机过程的薛定谔算子，称为随机薛定谔算子，在固体物理中起着重要的作用。首席研究员一直致力于IDS（状态的集成密度）和随机薛定谔算子的安德森定位问题。在与Klopp， Nakano和Nomura的联合论文中，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域性。在与Klopp的联合研究中，我们将类似的方法应用于所谓的随机跳跃模型来证明定位。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，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限的，因此我们不能得到波动方程理论中的传播定理。另一方面，已知初始状态的衰减意味着解的平滑性，这称为平滑效应。在一篇即将发表在《杜克数学》杂志上的论文中。J.)，证明了微局部平滑效应可以看作是（一类）波前集的传播，并将结果推广到具有长程型摄动主符号的薛定谔算子。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