Mathematical Analysis of Quantum Physics

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Various linear and non-linear partial differential equations and selfadjoint operators which appear in quantum physics have been investigated from wide range of spectrum by various methods. Principal results obtained in this period of research are the following1.The boundedness in Lebesgue spaces and Sobolev spaces of the wave operators of scattering theory for Schroedinger operators has been proved when spatial dimension is larger or equal to three including the case when the Schroedinger operator has the threshold singularities at the bottom of the continuous spectrum. The result is applied for the dispersive estimates for the solutions of the corresponding time dependent Schroedinmger equations2.For the solutions of time dependent Schroedinger equations the following results have been obtained. (1)The large time asymptotic formula is obtained for the first time in the case that the operator has the threshold singularities at the bottom of the continuous spectrum. (2)It is found and … More proved rigorously that the degree of the smoothing effect for the solutions is determined by the asymptotic behavior of the sojourn time in bounded sets of the corresponding classical particles when the energy becomes unbounded indefinitely. (3)For describing the propagation of micro-local singularities, the new notion of the homogeneous wave front sets is introduced and existing results on the propagation of analytic singularity or the singularities in the category of infinitely differentiability have been considerably improved, (4)It is found and proved rigorously that, in the semi-classical limit, solutions may be expanded in terms of resonances in the domains bounded by high wall of the potentials3.The extended phase method or the Floquet Hamiltonian method for studying time periodic system has been improved and, by using this, the asymptotic expansions of scattering solutions of time periodic Schroedinger equations have been obtained.4.The asymptotic completeness of scattering theory in the one photon sector of Nelson model, a simplified model of quantum electro-dynamics has been proved and the nature of the spectrum of the Hamiltonian in that sector has been determined.5.Mathematically rigorous definition of Feynman path integral has been given for the first time when the integrands are polynomially increasing functionals and the second term in the semi-classical expansion is obtained. Precise error estimates for the stationary phase method of oscillatory integrals in a large dimensional space have been obtained.6.The nature of the spectrum and the behavior of integrated density of states as the smoothness of Schroedinger operators with random potentials or random magnetic fields of specific types have been obtained. Less