2003 Fiscal Year Final Research Report Summary
OPERATOR-ANALYTICAL STUDY OF SINGULARITIES OF HAMILTONIANS IN QUANTUM PHYSICS
| Project/Area Number |
13640215
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Global analysis
|
|---|
| Research Institution | OKAYAMA UNIVERSITY |
|---|
Principal Investigator |
HIROKAWA Masao Okayama Univ., Fac. of Science, Associate Professor, 理学部, 助教授 (70282788)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
TAMURA Hideo Okayama Univ., Fac. of Science, Professor, 理学部, 教授 (30022734)
SATO Ryotaro Okayama Univ., Grad. Nat. Scien. Tech., Professor, 大学院・自然科学研究科, 教授 (50077913)
TANAKA Katsumi Okayama Univ., Fac. of Science, Associate Professor, 理学部, 助教授 (60207082)
HIROSHIMA Fumio Setsunan Univ., Fac. of Industry, Associate Professor, 工学部, 助教授 (00330358)
YAMADA Hirofumi Okayama Univ., Fac. of Science, Professor, 理学部, 教授 (40192794)
|
|---|
| Project Period (FY) |
2001 – 2003
|
| Keywords | nonrelativistic QED / infrared catastrophe / ultraviolet catastrophe / Pauli-Fierz model / Nelson model |
|---|
| Research Abstract |
We studied Nelson's model derived from the Pauli-Fierz model through several physical approximations. The Pauli-Fierz model describes an electron coupled with the quantized radiation field in nonrelativistic quantum electrodynamics, when we regard the electron as a nonrelativistic particle. We proved that the Nelson model has infrared catastrophe when its Hamiltonian has the Coulomb potential appearing in the structure of atoms. Developing the proof and using the Carleman operator, we clarified and characterized a mathematical mechanism which causes infrared safe or infrared divergence. The Carleman operator is derived from the so-called pull-through formula, and we gave the exact operator-theoretical proof for the formula, which is the fast to succeed in it. By this proof, we can investigate mathematical properties of the domain of the Carleman operator and pull-through formula, which resulted in our results. Because Nelson's model has infrared catastrophe by our results, we find another representation in which the model has a ground state. This representation describes the actual physical phenomenon. So, we removed both, infrared and ultraviolet cutoffs, and proved the Nelson model without both cutoffs has a ground state in the representation.
We studied the norm resolvent convergence for the Hamiltonian describing relativistic particle coupled with the Aharonov-Bohm field in 2-dim. space. We investigated which self-adjoint extension has the most suitable representation to the actual physics among several self-adjoint extensions corresponding to the boundary conditions around singularities.
|
Research Products
(10 results)
