| Co-Investigator(Kenkyū-buntansha) |
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)
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| Research Abstract |
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.
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