The Kac Transfer Operator, the Lie-Trotter Product Formula and Related Problems

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The research, primarily motivated by B.Helffer's work 1994-5 on the Kac transfer operator as well as Rogava's work in 1993 on the Lie-Trotter product formula in operator norm.(1) Ichinose and Takanobu considered the operator associated with the Levy process which is including the relativistic Schodinger operator, and proved (in Electronic J.Math.) by a probabilistic method based the Feynman-Kac formula sharp L^p-norm estimates between it semigroup and corresponding transfer operator.(2) Ichinose and Hideo Tamura has proved a very general result that the selfadjoint Lie-Trotter-Kato product formula holds in operator norm if the operator sum of two nonnegative selfadjoint operators is selfadjoint.(3) Hiroshi Tamura constructed an counterexample to support that our result in (2) is best possible.(4) Hideo Tamura keep studying with Hiroshi Ito the 2-demensional magnetic Schrodinger operator to watch the Aharanov-Bohm effect.(5) Yajima keeps studying how the fundamental solution of the Schrodinger equation behaves at infinity according as the potential is sub-quadratic or super-quadratic.