Project/Area Number  09440053 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
解析学

Research Institution  Kanazawa University 
Principal Investigator 
ICHINOSE Takashi Kanazawa University, Graduate School of Natural Science and Technology/Prof., 自然科学研究科, 教授 (20024044)

CoInvestigator(Kenkyūbuntansha) 
YAJIMA Kenji University of Tokyo, Graduate School of Mathematical Sciences/Prof., 大学院・数理科学研究科, 教授 (80011758)
TAMURA Hideo Okayama University, Faculty of Science/Prof., 理学部, 教授 (30022734)
NAKAO Shintaro Kanazawa University, Faculty of Science/Prof., 理学部, 教授 (90030783)
TAMURA Hiroshi Kanazawa University, Faculty of Science/Assoc.Prof., 理学部, 助教授 (80188440)
TAKANOBU Satoshi Kanazawa University, Faculty of Science/Assoc.Prof., 理学部, 助教授 (40197124)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥10,600,000 (Direct Cost : ¥10,600,000)
Fiscal Year 1998 : ¥4,600,000 (Direct Cost : ¥4,600,000)
Fiscal Year 1997 : ¥6,000,000 (Direct Cost : ¥6,000,000)

Keywords  transfer operator / Kac operator / LieTrotter product formula / Schrodinger operator / quantum mechanics / 転送作用素 / カッツ作用素 / リー・トロッター(LieTrotter)積公式 / シュレーディンガー作用素 / 量子力学 / シュレーデンガー作用素 / リ・トロッター(LieTrotter)積公 / シュレディンガー作用素 
Research Abstract 
The research, motivated by B.Helffer's work 19945 on the Kac transfen operator and Rogava's work 1993 on the LieTrotter 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 FeynmanKac 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 LieTrotter product formula in operator norm for Schrodinger operators. The results are obtained in both the nonrelativistic 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, operatortheoretical methods to prove almost the same nonrelativistic results as in (1). The LieTrotter 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 LieTrotter 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 LieTrotter product formula in operator norm that one of the recent results by NeidhardtZagrebnov gives an optimal error bound. He also obtained, with Keiichi 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.
