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

本研究的主要动机是B.Helffer 1994-5关于Kac转移算子的工作以及Rogava 1993年关于算子范数中的Lie-Trotter积公式的工作。(1) Ichinose和Takanobu考虑了与Levy过程相关的算子，其中包括相对论Schodinger算子，并利用基于Feynman-Kac公式的半群与相应转移算子之间的尖锐L^p范数估计的概率方法证明了（在《电子数学》中）。(2) Ichinose和Hideo Tamura证明了一个非常一般的结果，即当两个非负自伴随算子的算子和为自伴随时，自伴随Lie-Trotter-Kato积公式在算子范数下成立。(3) Hiroshi Tamura构造了一个反例来支持我们在(2)中的结果是最佳可能的。(4)田村秀夫与伊藤宏继续研究二维磁性薛定谔算子，观察Aharanov-Bohm效应。(5) Yajima继续研究薛定谔方程的基本解在无穷远处根据势是次二次或超二次时的表现。

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.