Project/Area Number  13640209 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Global analysis

Research Institution  Kyoto University 
Principal Investigator 
IWAI Toshihiro KYOTO UNIVERSITY Graduate School of Informatics, Prof., 情報学研究科, 教授 (10021635)

CoInvestigator(Kenkyūbuntansha) 
UWANO Yoshio KYOTO UNIVERSITY Graduate School of Informatics, Associate Prof., 情報学研究科, 助教授 (80201953)

Project Period (FY) 
2001 – 2003

Project Status 
Completed(Fiscal Year 2003)

Budget Amount *help 
¥3,400,000 (Direct Cost : ¥3,400,000)
Fiscal Year 2003 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 2002 : ¥1,200,000 (Direct Cost : ¥1,200,000)
Fiscal Year 2001 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  manybody system / reduction / stratification / 変換群論 / 力学系の層化簡約化 / 線形分子・非線形分子 / 量子力学系の層化・簡約化 / 多体系 / 力学系の対称性 / ピータ・ワイルの定理 
Research Abstract 
Geometric theory of dynamical systems in the title of this project is mainly concerned with manybody systems. If the collinear configurations are gotten rid of, the centerofmass system for many bodies is made into a principal fiber bundle. This allows us to set up reduction theory for manybody dynamical system in terms of connection. However, we showed that the restriction can be removed. In quantum mechanics, a key to a geometric reduction theory is PeterWeyl's theorem on unitary irreducible representations of compact Lie groups. By paying more attention to the rotation group and by applying the PeterWeyl theorem to wave functins on the centerofmass system, we were able to develop a quantum theory for manybody systems. The application of the PeterWeyl theorem is interpreted as a process of reduction by symmetry. In the course of this project, we have found that the theory of connections can be extended to be set up even if the structure group acts nonfreely. Thus we have established a stratified reduction theory for quantum manybody systems with rotational symmetry by stratifying the centerofmass system into stratum and by applying the PeterWeyl theorem to wave functions on each stratum. Actually, for quantum systems both for nonsingular configurtions and for collinear configurations, we have performed the reduction procedure by using rotational symmetry. We have also shown that the singulality of the kinetic energy operator at boundary of the main stratum does not cause the divergence of the energy integral.
