2000 Fiscal Year Final Research Report Summary
Spectral Theory for Schrodinger Operators with Magnetic Fields and its Application
| Project/Area Number |
11440056
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| Research Category |
Grant-in-Aid for Scientific Research (B).
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Okayama University |
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Principal Investigator |
TAMURA Hideo Okayama University, Science, Prof., 理学部, 教授 (30022734)
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| Co-Investigator(Kenkyū-buntansha) |
HIROKAWA Masao Okayama University, Science, Assistant Prof., 理学部, 助教授 (70282788)
KATSUDA Atsushi Okayama University, Science, Assistant Prof., 理学部, 助教授 (60183779)
SAKAI Takashi Okayama University, Science, Prof., 理学部, 教授 (70005809)
ITO Hiroshi Ehime University, Engineering, Assistant Prof., 工学部, 助教授 (90243005)
IWATSUKA Akira Kyoto Institute of Technology, Prof., 繊維学部, 教授 (40184890)
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| Project Period (FY) |
1999 – 2000
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| Keywords | magnetic Schrodinger operator / scattering at low energy / resolvent convergence / point interaction / scattering by magnetic field / Aharonov-Bohm effect |
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| Research Abstract |
The present project has been devoted to the study on the spectral and scattering theory for the Schrodinger operators with magnnetic fields. The special emphasis is placed on the mathematical study on the Aharonov-Bohm effect in magnetic scattering by point-like fields at large separation in two dimensions. The following three subjects has been studied.
(1) The asymptotic behavior at low energy of scattering amplitudes has been analysed for magnetic scattering in two dimensional fields, and the relation to scattering by magnetic fields with small support has been also discussed. The results obtained heavily depend on the flux of magnetic field and on the resonance space at zero energy.
(2) The Schrodinger operator with point-like magnetic field in two dimensions is known to be not essentially self-adjoint. It has the deficiency indices (2,2) and each self-adjoint extension is realized as a differential operator with some boundary conditions at the origin. We have studied which boundary condition is realized through the norm resolvent convergence to Schrodinger operator with point-like magnetic field when the support of magnetic fields shrinks.
(3) We have studied the Aharonov-Bohm effect in the scattering by two point-like magnetic fields at large separation. The asymptotic behavior of scattering amplitude has been analyzed when the distance between the centers of two fields goes to infinity. The obtained result heavily depends on the fluxes of fields and on incident and final directions.
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