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

本文在B.Helffer关于Kac Transfen算子的工作1994-5和Rogava关于算子范数下的Lie-Trotter乘积公式的基础上进行了研究，主要讨论了它与Schrodinger算子理论的联系：(1)Ichinose与Satoshi Takanobu用概率方法和Feynman-Kac公式证明了本项目标题中提到的势的估计比Helffer所讨论的更一般，并证明了Schrodinger算子的Lie-Trotter算子范数乘积公式。在非相对论和相对论两种情况下都得到了结果。数学课。太棒了。1997年，名古屋数学。J.1998)。此外，一篇论文正在准备中，它扩展到与Levy过程有关的更一般的算子的情况，包括相对论薛定谔算子。(2)Ichinose与田村秀夫和部分也与斗木使用算符理论方法来证明与(1)中几乎相同的非相对论结果。证明了Lie-Trotter乘积公式不仅是算子范数的，而且是迹范数的(J.Math.SoC。日本1998年，《渐近分析》1998)。对于算子范数下的Lie-Trotter乘积公式，在比Rogava公式更强的条件下，证明了比Rogava公式更好的误差界(Integr.平分。《理论》，大阪数学出版社1997年版。(3)田村弘(Hiroshi Tamura)在最近关于算子范数下的Lie-Trotter乘积公式的预印本中指出，Neidhardt-Zgrebnov最近的一个结果给出了一个最优误差界。他还与Kei-ichi Ito一起得到了O(N)Heisenberg模型临界温度上界的良好估计。(4)Yajima Kenji证明了薛定谔方程基本解的奇性。

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.