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2006 Fiscal Year Final Research Report Summary

Further Development of Trotter Product Formulas with Problems on Path Integrals

Research Project

Project/Area Number 16340038
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKanazawa University

Principal Investigator

ICHINOSE Takashi  Graduate School of Natural Science and Technology, Professor emeritus, 自然科学研究科, 名誉教授 (20024044)

Co-Investigator(Kenkyū-buntansha) TAMURA Hiroshi  Graduate School of Natural Science and Technology, Professor, 自然科学研究科, 教授 (80188440)
TAKANOBU Satoshi  Graduate School of Natural Science and Technology, Professor, 自然科学研究科, 教授 (40197124)
FUJIWARA Daisuke  Gakushuin University, Faculty of Science, Professor, 理学部, 教授 (10011561)
TAMURA Hideo  Okayama University, Graduate School of Natural Science and Technology, Professor, 自然科学研究科, 教授 (30022734)
YAJIMA Kenji  Gakushuin University, Faculty of Science, Professor, 理学部, 教授 (80011758)
Project Period (FY) 2004 – 2006
KeywordsTrotter product formula / Trotter-Kato product formula / Lie-Trotter-Kato product formula / exponential product formula / path integral / SchrOdinger operator / time-sliced approximation
Research Abstract

In two papers in Commun.Math.Phys. 2001, Ichinose, with Hideo Tamura, Hiroshi Tamura and V.A.Zagrebnov, proved norm convergence of Trotter product formula for the operator sum of two nonnegative selfadjoint operators with optimal error bound. The present research has been done to go beyond this result, keeping in mind its relation to path integral and examples/applications of the theory of Schrodinger operators, and brought the following results.
(1)Further development of Trotter product formula in norm-It is known that the norm convergence of Trotter product formula does not hold in general for the form sum of two selfadjoint operators. However, it was proved in J. Functional Analysis 2004 with some condition by Ichinose together with H. Neidhardt and V.A. Zagrebnov. It is an open problem whether the condition may be relaxed. To be unexpected,Ichinose and Hideo Tamura also discovered, in Lett. Math. Phys. 2004, norm convergence of the unitary product formula to hold for the Dirac and the relativistic SchrOdinger operator with nontrivial scalar potentials.
(2)Problems of path integral and of convergence of integral kernels of the Trotter product: Fujiwara succeeded to improve the error estimate of the stationary phase method of the oscillatory integral in large dimensions, and used it to determine the second term of the semiclassical approximation to Feynman path integral. In this connection, Trotter product formula is thought to give a kind of time-sliced approximation. The good norm-convergence of Trotter product formula might suggest convergence of the integral kernels, as Ichinose and Hideo Tamura anticipated. In fact, we proved it 2004 in two papers in Commn. PDE and J. Reine Angew. Math.
(3)Zeno product formula: Ichinose proved 2005 an intermediate result with P.Exner.

  • Research Products

    (12 results)

All 2006 2005 2004

All Journal Article (12 results)

  • [Journal Article] Exponential product approximation to integral kernel of Schrodinger semigroup and to heat kernel of Dirichlet Laplacian2006

    • Author(s)
      Takashi Ichinose, Hideo Tamura
    • Journal Title

      Journal fur die Reine und Angewandte Mathematik 592

      Pages: 157-188

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The second term of the semi-classical asymptotic expansion for Feynman path integrals with integrand of polynomial growth2006

    • Author(s)
      Daisuke Fujiwara, Naoto Kumano-go
    • Journal Title

      J. Math. Soc. Japan 58-3

      Pages: 837-867

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Exponential product approximation to integral kernel of Schrodinger semigroup and to heat kernel of Dirichlet Laplacian.2006

    • Author(s)
      Takashi Ichinose, Hideo Tamura
    • Journal Title

      Journal fur die Reine und Angewandte Mathematik Vol. 592

      Pages: 157-188

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The-second term of the semi-classical asymptotic expansion for Feynman path integrals with integrand of polynomial growth.2006

    • Author(s)
      Daisuke Fujiwara, Naoto Kumano-go
    • Journal Title

      J. Math. Soc. Japan Vol. 58-3

      Pages: 837-867

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A product formula related to quantum Zeno dynamics2005

    • Author(s)
      Pavel Exner, Takashi Ichinose
    • Journal Title

      Ann. H. Poincare 5

      Pages: 195-215

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A product formula related to quantum Zeno dynamics.2005

    • Author(s)
      Pavel Exner, Takashi Ichinose
    • Journal Title

      Ann. H. Poincare Vol.5

      Pages: 195-215

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Trotter-Kato product formula and fractional powers of self-adjoint generators2004

    • Author(s)
      Takashi Ichinose, Hagen Neidhardt, Valentin A.Zagrebnov
    • Journal Title

      J. Functional Analysis 207

      Pages: 33-57

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Note on the norm convergence of the unitary Trotter product formula2004

    • Author(s)
      Takashi Ichinose, Hideo Tamura
    • Journal Title

      Letters in Mathematical Physics 71

      Pages: 65-81

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Sharp error bound on norm convergence of exponential product formula and approximation to kernels of Schrodinger semigroups2004

    • Author(s)
      Takashi Ichinose, Hideo Tamura
    • Journal Title

      Comm. Partial Differential Equations 29

      Pages: 1905-1918

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Hagen Neidhardt and Valentin A. Zagrebnov, Trotter-Kato product formula and fractional powers of self-adjoint generators.2004

    • Author(s)
      Takashi Ichinose
    • Journal Title

      J. Functional Analysis Vol.207

      Pages: 33-57

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Note on the norm conv ergence of the unitary Trotter product formula.2004

    • Author(s)
      Takashi Ichinose, Hideo Tamura
    • Journal Title

      Letters in Mathematical Physics Vol.71

      Pages: 65・81

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Sharp error bound on norm convergence of exponential product formula and approximation to kernels of Schrodinger semigroups.2004

    • Author(s)
      Takashi Ichinose, Hideo Tamura
    • Journal Title

      Comm. Partial Differential Equations Vol.29

      Pages: 1905-1918

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2008-05-27  

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