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Spectral Theory for Schrodinger Operators with Magnetic Fields and its Application

Research Project

Project/Area Number 11440056
Research Category

Grant-in-Aid for Scientific Research (B).

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionOkayama University

Principal Investigator

TAMURA Hideo  Okayama University, Science, Prof., 理学部, 教授 (30022734)

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)
島田 伸一  摂南大学, 工学部, 助教授 (40196481)
Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥7,500,000 (Direct Cost: ¥7,500,000)
Fiscal Year 2000: ¥3,800,000 (Direct Cost: ¥3,800,000)
Fiscal Year 1999: ¥3,700,000 (Direct Cost: ¥3,700,000)
Keywordsmagnetic Schrodinger operator / scattering at low energy / resolvent convergence / point interaction / scattering by magnetic field / Aharonov-Bohm effect / スペクトル理論 / 磁場散乱 / アハロノフ・ボウム効果 / 散乱振幅
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.

Report

(3 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • Research Products

    (21 results)

All Other

All Publications (21 results)

  • [Publications] 田村英男: "Norm resolvent convergence to magnetic Schrodinger operators with point interactions"Rev.Math.Phys.. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 田村英男,伊藤宏: "Aharonov-Bohm effect in scattering by point-like magnetic fields at large separation"Ann.H.Poincare. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 田村英男,一瀬孝: "The norm convergence of the Trotter-Kato product formula with error bound"Commun.Math.Phys.. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 廣川真男: "Canonical quantization on a doubly connected space and the Aharonov-Bohm effect"J.Func.Anal.. 174. 322-363 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 岩塚明: "The uniqueness of the integrated density of states for the Schrodinger operators with magnetic fields"Math.Z.. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 田村英男: "Magnetic scattering at low energy in two dimensions"Nagoya Math.J.. 155. 95-151 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Tamura: "Magnetic scattering at low energy in two dimensions"Nagoya Math.J.. 155. 95-151 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Tamura: "Norm resolvent convergence to magnetic Schrodinger operators with point interactions"Rev.Math.Phys.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Tamura and H.T.Ito: "Aharonov-Bohm effect in scattering by point-like magnetic fields at large separation"Annales Henri Poincare. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Tamura and T.Ichinose: "The norm convergence of the Trotter-Kato product formula with error bound"Commun.Math.Phys.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Hirokawa: "Canonical quantization on a doubly connected space and the Aharanov-Bohm effect"J.Func.Anal.. 174. 322-363 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] A.Iwatsuka: "The uniqueness of the integrated density of states for the Schrodinger operators with magnetic fields"Math.Z.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 田村英男: "Norm resolvent convergence to magnetic Schrodinger operators with point interactions"Rev.Math.Phys.. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 田村英男: "Aharonov-Bohm effect in scattering by point-like magnetic fields at large separation"Ann.H.Poincare. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 田村英男: "The norm convergence of the Trotter-Kato product formula with error bound"Commun.Math.Phys.. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 廣川真男: "Canonical quantization on a doubly connected space and the Aharonov-Bohm effect"J.Func.Anal.. 174. 322-363 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 岩塚明: "The uniqueness of the integrated density of states for the Schrodinger operators with magnetic fields"Math.Z.. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 田村英男: "Magnetic scattering at low energy in two dimensions"Nagoya Math. J.. 155. 95-151 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 田村英男: "Asymptotic behavior of scattering amplitudes in magnetic fields at large separation"J. Math. Soc. Japan. (掲載予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 勝田 篤: "Isospectral graphs and isoperimetric constants"Discrete Math.. 207. 33-52 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 広川 真男: "An expression of the ground state energy of the spin-boson model"J. Func. Anul.. 162. 178-218 (1999)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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