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

Kac 转移算子与薛定谔半群之间差异的范数估计

基本信息

批准号：
09440053
负责人：
ICHINOSE Takashi
金额：
$ 6.78万
依托单位：
Kanazawa University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

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.

受B.Helffer（1994-5）关于Kac-Troten算子的工作和Rogava（1993）关于算子范数下的Lie-Trotter乘积公式的工作的启发，本文进行了这方面的研究，主要注意到它与Schrodinger算子理论的联系。(1)一之濑使用，与高信聪，概率方法与费曼-卡茨公式证明估计所提到的标题，本项目的潜力更普遍的比那些治疗的赫尔弗，也是李特罗特产品公式在运营商规范的薛定谔运营商。在非相对论和相对论两种情况下都得到了结果（Commun. Math.Phys.1997，名古屋Math.J.1998）。此外，一份文件是在准备的情况下，延伸到更一般的运营商与利维过程，包括相对论薛定谔运营商。(2)一之濑和田村秀夫，部分也和道目敦，用算子理论方法证明了与（1）中几乎相同的非相对论结果。Lie-Trotter乘积公式不仅在算子范数下而且在迹范数下都得到了证明（J. Math. Soc. Japan 1998，Asymptotic Analysis 1998）。对于算子范数下的Lie-Trotter乘积公式，虽然在比Rogava更强的条件下，但证明了比Rogava更好的误差界（Integr. Equat. Op. Theory 1997，Osaka J.Math.1998）。(3)田村浩在最近的预印本中指出，在算子范数下的李-特罗特乘积公式中，Neidhardt-Zagrebnov最近的一个结果给出了最优误差界。他还获得了，与伊藤圭一，良好的估计上限的临界温度的O（N）海森堡模型。(4)矢岛健二证明了薛定谔方程基本解的奇异性的一个非常尖锐的结果。