Further Development of Trotter-Kato Product Formula in Operator Norm and Related Problems on Path Integral

本文的研究主要是受B.Helffer在1994-5年关于Kac转移算子的工作以及Rogava在1993年关于算子范数下的自伴Trotter-Kato乘积公式的工作的启发。（1）Ichinose和Hideo Tamura证明了一个非常一般的结果：当两个非负自伴算子的算子和是自伴的时，自伴Trotter-Kato乘积公式在算子范数下成立，并且Hiroshi Tamura和V. A. Zagrebnov证明了这个结果的最优误差界为O（1/n）。（2）在形式和情形下，Ichinose与H.Neidhardt和V. A. Zagrebnov证明了当其中一个算子相对于另一个算子是形式有界的，且在这些算子的定义域上附加了一些条件时，在算子范数下具有最优误差界的自伴Trotter-Kato乘积公式。（3）田村秀夫还与伊藤弘研究了二维磁薛定谔算子，通过散射振幅的渐近行为观察Aharanov-Bohm效应。（4）Yajima研究了非相对论量子电动力学中时间周期薛定谔方程和纳尔逊模型解的局域时-dcay。

The research is primarily motivated by B.Helffer's work 1994-5 on the Kac transfer operator as well as Rogava's work in 1993 on the selfadjoint Trotter-Kato product formula in operator norm.(1)Ichinose and Hideo Tamura proved a very general result to the effect that the selfadjoint Trotter-Kato product formula holds in operator norm if the operator sum of two nonnegative selfadjoint operators is selfadjoint, and also, with Hiroshi Tamura and V.A.Zagrebnov, this result with optimal error boundO(1/n).(2)Further, for the form sum case, Ichinose proved, with H.Neidhardt and V.A.Zagrebnov, the selfadjoint Trotter-Kato product formula in operator norm with optial error bound, in case where one of these operators is form-bounded with respect to the other, with some additional condition on the domains of these operators.(3)Hideo Tamura also studied with Hiroshi Ito the 2-dimensional magnetic Schrodinger operator to watch the Aharanov-Bohm effect through the asymptotic behaviour of the scattering amplitude.(4)Yajima studied local time-dcay of solutions of the time-periodic Schrodinger equation and the Nelson model in nonrelativistic quantum electrodynamics.