Spectral and Scattering Theory for Schrodinger Equations

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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