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The standard realization of crystal lattices and spectra of magnetic transition operators

Research Project

Project/Area Number 14540057
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTohoku University

Principal Investigator

KOTANI Motoko  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50230024)

Co-Investigator(Kenkyū-buntansha) FUJIWARA Koji  Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (60229078)
SHIOYA Takashi  Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90235507)
SUNADA Toshikazu  Meiji University, School of Science and Technology, Professor, 理工学部, 教授 (20022741)
OHNITA Yoshihiro  Tokyo Metropolitan University, Faculty of Science, Professor, 大学院・理学研究科, 教授 (90183764)
IZEKI Hiroyasu  Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90244409)
納谷 信  名古屋大学, 大学院・多元数理科学研究科, 助教授 (70222180)
Project Period (FY) 2002 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2003: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2002: ¥1,500,000 (Direct Cost: ¥1,500,000)
Keywordscrystal lattice / magnetic transition operators / central limit theorem / ランダムウォーク / ハーパー作用素
Research Abstract

A crystal lattice is an abelian covering infinite graph of a finite graph. The integer lattices, the triangular lattice, the hexagonal lattice are examples of crystal lattices. We define magnetic transition operators to describe electron transfer on a crystal lattice under periodic magnetic field. The definition is justified by the central limit theorem : Namely, we show that the semigroup generated by the magnetic transition operators converges to the semigroup generated by a magnetic Laplacian of the Euclidean space with the Albanese metric. Magnetic fields on a crystal lattice are defined in terms of the second group cohomology. Next we construct a C^*-algebra associated with the magnetic field and show the magnetic transition operator belongs to the C^*-algebra. By using this, we show the spectra of the magnetic transition operators is a Lipschitz continuous function in magnetic field.
Without magnetic field, electrons behave like random walks. We show large deviation principle holds for random walks on a crystal lattice. By letting lattice spacing smaller, a crystal lattice converges to a finite dimensional vector space with a Banach norm in the Gromov-Hausdorff topology. This Banach norm is characterized in terms of the rate function appearing in the large deviation.

Report

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

    (32 results)

All Other

All Publications (32 results)

  • [Publications] M.Kotani: "Asymptotic of Large deviation for random walks on a crystal lattice"Contemporary Math.. (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Kotani, T.Sunada: "Spectral geometry of crystal lattice"Contemporary Math.. (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Kotani: "Lipschitz continuity of the spectra of the magnetic transition operators on crystal lattice"J.Geom and Phys.. 47. 323-342 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Kotani: "A central limit theorem for magnetic transition operators on a crystal lattice"J.London Math.Soc.. 65. 464-482 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Kuwae, T.Shioya: "Sobolev and Dirichlet spaces over maps between metric spaces"J.Reine Angew Math.. 555. 39-75 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Fujiwara: "On the outer automorphism group of a hyperbolic group"Israel J of Math. 131. 277-284 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Kotani: "an asymptotic of the large deviation for random walks on a crystal lattice"Contemporary Math.. (to appear). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Kotani, T.Sunada: "Spectral geometry of crystal lattices"Contemporary Math.. (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Kotani: "Lipscitz continuity of the spectra of the magnetic transition operators a crystal lattice"J.Geom.Phys.. 47. 323-342 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Kotani: "A central limit theorem for magnetic transition operators on a crystal lattice"J.London Math.Soc.. 65. 464-482 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Kuwae, T.Shioya: "Sobolev and Dirichlet spaces over maps between metric spaces"J.Reine Angew.Math.. 555. 39-75 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Fujiwara: "On the outer automorphism group of a hyperbolic group"Israel J of Math. 131. 27-284 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Kotani: "Asymptotic of Large deviation for random walks on a crystal lattice"Contemporary Math.. (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Kotani, T.Sunada: "Spectral geometry of crystal lattice"Contemporary Math.. (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Kotani: "Lipschitz continuity of the spectra of the magnetic transition operators on crystal lattice"J.Geom and Phys.. 47. 323-342 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Kotani: "A central limit theorem for magnetic transition operators on a crystal lattice"J.London Math.Soc.. 65. 464-482 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Kuwae, T.Shioya: "Sobolev and Dirichiet spaces over maps between metric spaces"J.Reine Angew Math.. 555. 39-75 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Motoko Kotani: "A note on asymptotic expansions for closed geodesics in homology classes"Matn. Ann.. 320. 507-529 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Kotani, T.Sunada: "The pressure and higher correlations for an Anosov diffeomorphism"Ergod. Th. Dynam. Sys.. 21. 807-821 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Kotani: "A central limit theorem for magnetic transition operators on a crystal lattice"J. London Math. Soc.. 65. 464-482 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Kotani: "Lipscitz continuity of the spectra of the magnetic transition operators on a crystal lattice"J. Geom and Phys.. (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Fujiwara, K.Ohshika: "The second bounded cohomology of 3-manifolds"Publ. Res. Inst. Math. Sci.. 38. 347-354 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Fujiwara, T.Soma: "Bounded classes in the cohomology of manifolds"Geom. Dedicata. 92. 73-85 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Bestvina, K.Fujiwara: "Bounded cohomology of subgroups of mapping class groups"Geom. and Topology. 6. 69-89 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Fujiwara: "On the outer automorphism group of a hyperbolic group"Israel J. of Math.. 131. 277-284 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y Ohnita, S. Udagawa: "Harmonic maps of finite type into generalized flag manifolds and twistor fibrations"Contem Math.. 308. (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Amarzaya, Y.Ohnita: "Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces"Tohoku Math. J..

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Kuwae, T.Shioya: "Convergence of spectral structures : a functional analytic theory and its applications to spectral geometry"Comm. in Anal. and Geom..

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Kuwar, T.Shioya: "Sobolev and Dirichlet spaces over maps between metric spaces"J. Reine and Angewandle Math..

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Shioya: "Behavior of distant maximal geodesics in finitely connected 2-dimensional Riemannian manifolds II."Geom. Dedicata.

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Izeki: "Convex-cocompactness of Kleinian groups and conformally flat manifolds with positive scalar curvature"Proc. Amer. Math. Soc.. 130. 3731-3740 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Kamada, S.Nayatani: "Quaternionic analogue of CR geometry"Semm. Theor. Spectr. Geom.. 19. 41-52 (2001)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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