Project/Area Number  07640243 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  Gakushuin University, Faculty of Science 
Principal Investigator 
KURODA Shigetoshi Gakushuin Univ., Dept.of Mathematics, Professor, 理学部, 教授 (20011463)

CoInvestigator(Kenkyūbuntansha) 
MITSUI Takayoshi Gakushuin Univ., Dept.of Mathematics, Professor, 理学部, 教授 (20080484)
IITAKA Shigeru Gakushuin Univ., Dept.of Mathematics, Professor, 理学部, 教授 (20011588)
KATASE Kiyoshi Gakushuin Univ., Dept.of Mathematics, Professor, 理学部, 教授 (70080489)
MIZUTANI Akira Gakushuin Univ., Dept.of Mathematics, Professor, 理学部, 教授 (80011716)
FUJIWARA Daisuke Gakushuin Univ., Dept.of Mathematics, Professor, 理学部, 教授 (10011561)

Project Fiscal Year 
1995 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥2,000,000 (Direct Cost : ¥2,000,000)
Fiscal Year 1996 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1995 : ¥1,300,000 (Direct Cost : ¥1,300,000)

Keywords  Schrodinger equation / Spectral theory / Semilinear elliptic boundary value problem / Feynmann integral / Pass integral / Degenerate nonlinear parabolic equations / Explicit method / Finite element approximation / シュレディンガー方程式 / スペクトル理論 / ファインマン積分 / 経路積分 / 準線形線形楕円形境界値問題 / 退化放物型方程式 / 有限要素近似 / 陽解法 / 漸近解析 / 振動積分 / 有限要素法近似 / マンデルブロ集合 
Research Abstract 
1. Spectral Theory. A substaintial generalization of the distributional rankone perturbation theory due to B.Simon et al was made. Systematic applications to point and surface interactions will be the subject of future investigation. Another topics of research was the theory of eigenvalue distribution of modified scattering matrix due to R.D.Yafaev and it is shown that the theoryis applicable to problems with nonspherically symmetric vector potentials. 2. Pass Integfral and asymptotic analysis. Continuing previous research, it was shown that timeslicing approximations converge to the path integral in the strong topology. A simple and direct new proof of theorems of KumanogoTaniguchi type was given. 3. Numerical analysis of Schrodinger equations. Finite step explicit method for timedependent Schrodinger equation was investigated from a new viewpoint and an algorithn for nonlinear Schroodinger equation was proposed. 4. Semilinear elliptic boundary value problems. It was shown that based on the Nehari variational principle some kind of unstable solutions can be constructed in numerically efficient ways and the effectiveness of the method was confirmed by numeraicl experiments. 5. Degenerate nonlinear parabolic equations. Finite difference approximations were investigated and for some specific problems the convergence of approximate solution to the true solution was proved. Numerical experiments were also performed. 6. Real analysis method. As applications of real analysis methods some problems of weighted estimates were obtained for the inverse of the Schrodinger operator with certain potentials. In the field of classical real analysis some norm estimates of Kakeya's maximal function were obtained for functions with separable variables. 7. Complex dynamical system. New knowledge were obtained about the periodicity of heperbolic points of the Mandelbrot set and the behavior of the dynamical system around Misiurewicz points.
