1999 Fiscal Year Final Research Report Summary
Mathematics of Schrodinger Equations
| Project/Area Number |
10640186
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | Gakushuin University |
|---|
Principal Investigator |
KURODA Shige toshi Gakushuin Univ., Dept.of Mathematics, Professor, 理学部, 教授 (20011463)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
JIN Naondo Gakushuin Univ., Dept.of Mathematics, Research Associate, 理学部, 助手 (90206368)
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)
KURATA Kuzuhiro Tokyo Metrop. Univ., Dept.of Mathematics, Associate Professor, 理学部, 助教授 (10186489)
SUZUKI Toshio Yamanashi Univ., Dept.of Math.and Phys., Professor, 教育人間科学部, 教授 (20020472)
|
|---|
| Project Period (FY) |
1998 – 1999
|
| Keywords | Schrodinger equation / Spectral theory / Feynman integral / Stationary phase method / Zeros of Polynomials / Nonlinear PDE / Covering surface of the unit disc |
|---|
| Research Abstract |
1. Singular interaction and spectral theory. Being motivated by the construction of the Hamiltonian for schrodinger equations with singular potential like point interactions, a new method of constructing a perturbed Hamiltonian through a generalized resolvent equation was developed. A clue was obtained to relate this method to the Krein formula.
2. More singular interraction was considered in problem 1 and it was found that in a certain case the Hamiltonian can be defined as a selfadjoint operator in a bigger Hilbert space.
3. Feynman Integral. Fujiwara's research on the mathematical theory for Feynman integrals was comprehensively presented in a book form. The research on the stationary phase method for large number of dimension was continued.
4. Degenerate parabolic equations. LィイD11ィエD1 convergence of the finite-element approximation was shown together with some numerical verification.
5. Covering surfaces. Two-sheeted covering surfaces of the unit disc was investigated and it was shown that a sufficient condition for the maximality is expressed in terms of the Kuramochi boundary.
6. Schrodinger equations with magnetic potential. Many results were obtained on schrodinger operators with a magnetic potential, such as the low energy scattering (two space dimension) and the asymptotics in the discrete spectrum of Pauli operators.
7. Zeros of polynomials. A remarkable new algorithm was obtained for the mumerical computation of zeros of a polynomial. Implementability of the algorithm was also tested.
8. 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.
9. Nonlinear partial differential equations. Various results were obtained for linear and nonlinear partial differential equations. The methods include real analytic one.
|
