Project/Area Number  08454042 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  KYOTO UNIVERSITY 
Principal Investigator 
TAKAHASHI Yoichiro RIMS,KYOTO UNIVERSITY Professor, 数理解析研究所, 教授 (20033889)

CoInvestigator(Kenkyūbuntansha) 
OHKITANI Koji RIMS,KYOTO UNIVERSITY Assistant Professor, 数理解析研究所, 助教授 (70211787)
TAKEI Yoshitsugu RIMS,KYOTO UNIVERSITY Assistant Professor, 数理解析研究所, 助教授 (00212019)
KAWAI Takahiro RIMS,KYOTO UNIVERSITY Professor, 数理解析研究所, 教授 (20027379)
MUROTA Kazuo RIMS,KYOTO UNIVERSITY Professor, 数理解析研究所, 教授 (50134466)
OKAMOTO Hisashi RIMS,KYOTO UNIVERSITY Professor, 数理解析研究所, 教授 (40143359)

Project Fiscal Year 
1996 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥5,200,000 (Direct Cost : ¥5,200,000)
Fiscal Year 1996 : ¥5,200,000 (Direct Cost : ¥5,200,000)

Keywords  random matrix / chaos / hydrodynamical limit / asymptotic theory / singular perturbation / eigenvalue distirbution / point process / classical quantum correspondence / ランダム行列 / カオス / 流体力学的極限 / 漸近理論 / 特異摂動 / 固有値分布 / 点過程 / 古典量子対応 
Research Abstract 
Quantum chaos or the classicalsemiiclassicalquantum correspondence is one of the main topics around the problem of the chaos of dynamical systems. The purpose of the research was to investigate the mathematical problems in the following three directions. 1) Discrete analogue and 2) Trace formulas. One of the implications carried by the concept of LiYorke's chaos is clarified by Y.Takahashi (with Y.Baba and I.Kubo). T.Shirai obtained a remarkable result : a graph version of Gutzwiller's formula does hold exactly. 3) Hydrodynamical limit of many particle systems and analysis of the limit equations. There are several results in this direction. K.Ohkitani computed Lyapunov exponents for the developped turbulence (with Yamada), H.Okamoto and Sakajo studied the rollup phenomenon of vortex sheet, K.Murota discussed the bifurcation of soil shear. Also, the exact theory of singular perturbation due to H.Kawai and Y.Takei is suggesive in this direction. Y.Takahashi organized a meeting on asymptotic theory in stochastic analysis, including lectures concerning the recent development of oscillatory integral theory. He also gave a new simpler proof to the result of 16 years ago (with K.Hara) and tried to describe the results in random matrix theory from the viewpoint of point process and its scaling limit (partially, announced at Oberwolfach institute). Also, he started the stochasticanalytical study of the quadratic form associated with the Cauch singular integral inspired by a result of Johansson (1995) extending Szego's result in finite dimension (with T.Chiyonobu).
