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OPERATOR-ANALYTICAL STUDY OF SINGULARITIES OF HAMILTONIANS IN QUANTUM PHYSICS

Research Project

Project/Area Number 13640215
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionOKAYAMA 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
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2003: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2002: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2001: ¥600,000 (Direct Cost: ¥600,000)
Keywordsnonrelativistic QED / infrared catastrophe / ultraviolet catastrophe / Pauli-Fierz model / Nelson model / パウリ・フイエルツ模型 / 場の量子論 / Nelson Model / spin-boson model / Wigner-Weisskoph model / Pauli-Fierz 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.

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] M.Hirokawa: "Ground State for Point Interacting through a Massless Scalar Bose Field"Advances in Mathematics. (accepted).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] J.Dittrich: "A Model of Interband Radiative Transition"Journal of the Mathematical Society of Japan. 56(in press). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Hiroshima: "Localization of the Number of Photons of Ground States in Nonrelativistic QED"Reviews in Mathematical Physics. 15・3. 271-312 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Tamura: "Resolvent Convergence in Norm or Dirac Operator with Aharonov-Bohm Field"Journal of Mathemaical Physics. 44・7. 2967-2993 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Hirokawa: "A Garden of Quanta. Essays in Honor of Hiroshi Ezawa, 209 - 242"World Scientific. 502 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Hirokawa: "Ground State for Point Interacting through a Massless Scalar Bose Field"Advances in Mathematics. (accepted).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] J.Dittrich: "A Model of Interband Radiative Transition"Journal of the Mathematical Society of Japan. vol.56(in press). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Hiroshima: "Localization of the Number of Photons of Ground States in Nonrelativistic QED"Reviews in Mathematical Physics. vol.15(3). 271-312 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Tamura: "Resolvent Convergence in Norm for Dirac Operator with Aharonov-Bohm Field"Journal of Mathematical Physics. vol.44(7). 2967-2993 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Hirokawa: "Recent Developments in Mathematical Methods for Models in Non-Relativistic Quantum Electrodynamics"A Garden of Quanta. Essays in Honor of Hiroshi Ezawa (World Scientific). 209-242 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Hirokawa: "Ground State for Point Interacting through a Massless Scalar Bose Field"Advances in Mathematics. (accepted).

    • Related Report
      2003 Annual Research Report
  • [Publications] J.Dittrich: "A Model of Interband Radiative Transition"Journal of the Mathematical Society of Japan. 56(in press). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] F.Hiroshima: "Localization of the number of photons of ground states in nonrelativistic QED"Reviews in Mathematical Physics. 15・3. 271-312 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Tamura: "Resolvent convergence in norm for Dirac operator with Aharonov-Bohm field"Journal of Mathematical Physics. 44・7. 2967-2993 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Hirokawa: "A Garden of Quanta. Essays in Hornor of Hiroshi Ezawa,209--242"World Scientific. 502 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Hirokawa: "Ground state transition for two-level system coupled with Bose field"Phys. Lett. A. 294. 13-18 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] F.Hiroshima: "Embedded eigenvalues, localization and asymptotics of quantum field model : functional integral annroach"J. Phys. A. 35. 351-375 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] F.Hiroshima: "Observable effects and parametrized scaling limits of a model in nonrelativistic quantum electrodynamics"J. Math. Phys.. 43. 1755-1795 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Ikeda, H.Yamada: "Polynomial τ-functions of the NLS-Toda hierarchy and the Virasoro singular vectors"Lett. Math. Phys.. 60. 147-156 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Hirokawa: "Remarks on Ground State Energy of the Spin-Boson Model. An Application of the Winer-Weisskoof Model"Reviews in Mathematical Physics. 13・2. 221-251 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] A.Arai, M.Hirokawa: "Stability of Ground States in Sectors and Its Application to the Wigner-Weisskopf Model"Reviews in Mathematical Physics. 13・4. 513-527 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Hirokawa, 0.Ogurisu: "Ground State of a Spin-1/2 Charged Particle in a Two-Dimensional Magnetic Field"Journal of Mathematical Physics. 42・8. 3334-3343 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] F.Hiroshima: "Ground states and spectrum of quantum electrodynamics of nonrelativistic particles"Trans.Amer.Math.Soc.. 353・11. 4497-4528 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.T.Ito, H.Tamura: "Aharonov-Bohm effect in scattering by point-like magnetic fields at large separation"Ann.Henri Poincare. 2・2. 309-359 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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