The norm estimate of the difference between the Kac transfer operator and the Schrodinger semigroup

在B.Helffer 1994-5关于Kac转移算子的研究和Rogava 1993关于算子范数中的Lie-Trotter积公式的研究的推动下，本文进行了研究，主要注意其与薛定谔算子理论的联系。(1) Ichinose与Satoshi Takanobu使用概率方法和Feynman-Kac公式证明了本课题标题中提到的对于比Helffer处理的更一般的势的估计，以及对于薛定谔算子算子范数中的Lie-Trotter积公式。结果可以在非相对活动和相对论情况下得到。数学。物理，1997，名古屋数学。J.1998)。此外，一篇论文正在准备中，它扩展到与Levy过程相关的更一般的算符的情况，包括相对论性薛定谔算符。(2) Ichinose与田村秀夫（Hideo Tamura）以及部分与Atsushi Doumeki一起使用算子理论方法证明了与(1)中几乎相同的非相对论性结果。李氏积公式不仅在算子范数上得到了证明，而且在迹范数上得到了证明。Soc。日本1998，渐近分析1998)。对于算子范数中的Lie-Trotter积公式，证明了一个比Rogava的更好的误差界，但条件比他的更强。Equat。Op. Theory, 1997，大阪J.Math。1998)。(3) Hiroshi Tamura在最近的一篇关于算子范数中的Lie-Trotter积公式的预印本中指出Neidhardt-Zagrebnov最近的一个结果给出了最优误差界。他还与伊藤庆一（Kei-ichi Ito）一起，对O(N)海森堡模型的临界温度上界进行了很好的估计。(4)矢岛健二证明了薛定谔方程基本解奇异性的一个非常尖锐的结果。

The research, motivated by B.Helffer's work 1994-5 on the Kac transfen operator and Rogava's work 1993 on the Lie-Trotter product formula in operator norm, has been carried out, mainly noting its connection with the theory of Schrodinger operators.(1) Ichinose used, with Satoshi Takanobu, probabilistic methods with the Feynman-Kac formula to prove the estimates as mentioned in the title of this project for potentials more general than those treated by Helffer, and also the Lie-Trotter product formula in operator norm for Schrodinger operators. The results are obtained in both the nonrel-ativistic and relativistic cases (Commun. Math. Phys. 1997, Nagoya Math. J.1998). Further, a paper is in preparation which extentds to the case of the more general operators associated with the Levy process including the relativistic Schrodinger operator.(2) Ichinose used, with Hideo Tamura and partly also with Atsushi Doumeki, operator-theoretical methods to prove almost the same nonrelativistic results as in (1). The Lie-Trotter product formulas were also proved not only in operator norm but also in trace norm (J.Math. Soc. Japan 1998, Asymptotic Analysis 1998). As for the Lie-Trotter product formula in operator norm, a better error bound than Rogava's, though under a stronger condition than his, was proved (Integr. Equat. Op. Theory 1997, Osaka J.Math. 1998).(3) Hiroshi Tamura has noted in a recent preprint on the Lie-Trotter product formula in operator norm that one of the recent results by Neidhardt-Zagrebnov gives an optimal error bound. He also obtained, with Kei-ichi Ito, good estimates for the upper bound of the critical temperature of O(N) Heisenberg models.(4) Kenji Yajima proved a very sharp result on the singularity of the fundamental solution for the Schrodinger equation.