2000 Fiscal Year Final Research Report Summary
Spectral Theory for Schrodinger Operators with Magnetic Fields and its Application
Project/Area Number 
11440056

Research Category 
GrantinAid for Scientific Research (B).

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Global analysis

Research Institution  Okayama University 
Principal Investigator 
TAMURA Hideo Okayama University, Science, Prof. > 岡山大学, 理学部, 教授 (30022734)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Period (FY) 
1999 – 2000

Keywords  magnetic Schrodinger operator / scattering at low energy / resolvent convergence / point interaction / scattering by magnetic field / AharonovBohm effect 
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 AharonovBohm effect in magnetic scattering by pointlike 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 pointlike magnetic field in two dimensions is known to be not essentially selfadjoint. It has the deficiency indices (2,2) and each selfadjoint 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 pointlike magnetic field when the support of magnetic fields shrinks. (3) We have studied the AharonovBohm effect in the scattering by two pointlike 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.

Research Products
(12 results)