Study of asymptotic distribution of the Schroedinger operators with strong magnetic fields

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Iwasuka showed that the spectrum of the Schroedinger operators with Poisson random potential is the whole real line potential has some negative part.Mine obtained some detailed estimate of the number of the eigenvalues lying between the Landau levels when constant magnetic field is perturbed by multiple solenoidal (point) magnetic field, by using method of the perturbation of the canonical commutation relation.Doi studied the structure of the singularity of the solution to the Schroedinger equations with quadratic potential with perturbation by using the analysis of the asymptotic behavior of the Hamilton flow.Ito and Tamura proved the selfadjointness, the asymptotic completeness of the wave operator and the nonexistence of the eigenvalue for the Schrodinger operators with multiple delta like magnetic fields, and gave the leading term of the asymptotic behavior of their scattering amplitude when the distance between the centers of the magnetic fields tends to infinity.Tamura also proved the convergence in norm of the resolvent of the two dimensional Dirac operators when a delta like magnetic field is approximated by smooth fields. He also obtained an asymptotic formula for the kernel of the Schroedinger semigroup in the case of singular potentials by applying his recent result that the Trotter product formula converges in the sense of operator norm for unbounded operators.