1996 Fiscal Year Final Research Report Summary
Functional Analysis and Partial Differential Equations
| Project/Area Number |
07640243
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| 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)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
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 Period (FY) |
1995 – 1996
|
| Keywords | Schrodinger equation / Spectral theory / Semi-linear elliptic boundary value problem / Feynmann integral / Pass integral / Degenerate non-linear parabolic equations / Explicit method / Finite element approximation |
|---|
| Research Abstract |
1. Spectral Theory. A substaintial generalization of the distributional rank-one 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 non-spherically symmetric vector potentials.
2. Pass Integfral and asymptotic analysis. Continuing previous research, it was shown that time-slicing approximations converge to the path integral in the strong topology. A simple and direct new proof of theorems of Kumano-go-Taniguchi type was given.
3. Numerical analysis of Schrodinger equations. Finite step explicit method for time-dependent Schrodinger equation was investigated from a new viewpoint and an algorithn for non-linear Schroodinger equation was proposed.
4. Semi-linear 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 non-linear 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.
|
