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Spectra of Elliptic Operators on Manifolds and Classical Mechanics

Research Project

Project/Area Number 11640205
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionThe University of Tokushima

Principal Investigator

KUWABARA Ruishi  The University of Tokushima, Faculty of Integrated Arts and Sciences Professor, 総合科学部, 教授 (90127077)

Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2000: ¥500,000 (Direct Cost: ¥500,000)
KeywordsHamiltonian dynamical system / Schrodinger operator / Laplacian / Spectrum / Periodic orbit / Quantization condition / Fourier integral operator / Magnetic field / グラフ / 閉測地線 / 接続のホロノミー
Research Abstract

The purpose of the research project is to investigate the relationships between the properties of classical mechanics and the spectrum of the associated Schrodinger operator on the Riemannian manifolds.
Particularly, we have payed attention to the mechanics in a magnetic field on the Riemannian manifold. A magnetic field is regarded as a closed two-form on the manifold, and the motion of a charged particle in the magnetic field is formulated as the flow of the Hamiltonian system with the symplectic structure twisted by the two-form. On the other hand, the associated quantum system or the Schrodinger operator is the Laplacian on the complex line bundle naturally defined by the integral closed two-form (the magnetic field) on the manifold. In this context, we have obtained the following results :
1. We have considered the quantization condition for the invariant torus of the Hamiltonian system of magnetic flow, and have clarified by virtue of the theory of Fourier integral operators that the (semi-classical) energy levels determined by the quantization condition give a "good" approximation of the true energies of the quantum system.
2. We have, moreover, considered the "quantization condition" for the stable periodic orbits of the magnetic flow, and have similarly clarified that the classical "energy" of the a suitable periodic orbit gives an approximation of the energy of the associated quantum system. The tool used in this research if the theory of Fourier integral operators of Hermite type which are the operator corresponding to the isotropic submanifold.

Report

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

    (9 results)

All Other

All Publications (9 results)

  • [Publications] 桑原類史: "力学系の古典軌道と量子エネルギー分布"数理解析研究所講究録. 1119巻. 26-34 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Ruishi Kuwabara: "On Maslov's quantization condition for mechanics in a magnetic field"J.Math.Tokushima Univ.. 33巻. 33-54 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 桑原類史: "磁場における力学系の周期軌道と量子エネルギー分布"数理解析研究所講究録. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Ruishi Kuwabara: "Classical orbits and quantum energies of mechanics (in Japanese)"RIMS.Kokyuroku. vol.1119. 26-34 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Ruishi Kuwabara: "On Maslov's quantization condition for mechanics in a magnetic field"J.Math.Tokushima Univ.. vol.33. 33-54 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Ruishi Kuwabara: "Periodic orbits and quantum energies of mechanics in a magnetic field (in Japanese)"RIMS.Kokyuroku. (in Press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 桑原類史: "磁場における力学系の周期軌道と量子エネルギー分布"数理解析研究所講究録. (印刷中).

    • Related Report
      2000 Annual Research Report
  • [Publications] Ruishi Kuwabara: "On Maslov's quantization condition for mechnics in a magnetic field"J.Math.Tokushima Univ.. 33. 33-54 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Ruishi Kuwabara: "Difference spectrum of the Schrodinger operator in a magnetic field"Math.Zeit.. (印刷中).

    • Related Report
      1999 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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