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

Kac 转移算子、Lie-Trotter 乘积公式及相关问题

• 批准号: 11440040
• 负责人: ICHINOSE Takashi
• 金额: $ 3.84万
• 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: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440040/
• 关键词: transfer operator; Kac operator; Lie-Trotter product formula; Schrodinger operator; quantum mechanics

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.

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Lie-Trotter乘积公式的工作的启发。(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，并证明了（在Electronic J.Math.）利用基于Feynman-Kac公式的概率方法，得到了它的半群与相应的转移算子之间的L^p-模的精确估计. (2)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Lie-Trotter-Kato乘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)田村秀夫继续与伊藤弘一起研究二维磁薛定谔算符，观察Aharanov-Bohm效应。(5)矢岛一直在研究薛定谔方程的基本解在无穷远处的行为，这取决于势是次二次势还是超二次势。

项目成果

Takashi Ichinose: "The norm estimate of the difference between the Kac operator and the Schrodinger semigroup II : The general case including the relativistic case"Electronic Journal of Probability. 5. 1-47 (2000)

Takashi Ichinose：“Kac 算子与薛定谔半群 II 之间差异的范数估计：包括相对论情况的一般情况”《电子概率杂志》。

Takashi ICHINOSE: "The norm estimate of the difference between the Kac operator and Schroedinger semigroup ll: The general case including the relativistic case"To appear in Electronic Journal of Probability. (2000)

Takashi ICHINOSE：“Kac 算子和薛定谔半群 ll 之间差异的范数估计：包括相对论情况的一般情况”出现在《电子概率杂志》上。

Keiichi ITO: "N dependence of upper bounds of critical temperatures of 2D O(N) models"Commun. Math. Phys.. 202. 127-168 (1999)

Keiichi ITO：“2D O(N) 模型临界温度上限的 N 依赖性”Commun。

Takashi Ichinose and Hideo Tamura: "The norm convergence of the Trotter-Kato product formula with error bound"to appear in Commun.Math.Phys..

Takashi Ichinose 和 Hideo Tamura：“具有误差界的 Trotter-Kato 乘积公式的范数收敛性”出现在 Commun.Math.Phys..

Kenji Yajima: "On the behavior at infinity of the fundamental solution of the time dependent Schrodinger equations"to appear in Reviews in Math.Phys..

Kenji Yajima：“关于与时间相关的薛定谔方程的基本解的无穷远行为”出现在数学物理评论中。