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On the spectrum of the one-dimensional Schitdinger operators with periodic point -interactions

Research Project

Project/Area Number 18540190
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

YOSHITOMI Kazushi  Tokyo Metropolitan University, Graduate School of Science and Engineering, Associate Professor (40304729)

Project Period (FY) 2006 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,880,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥480,000)
Fiscal Year 2007: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2006: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsPoint-interactions / Periodic potentials / SchrOdinger operators / Spectral gaps / Asymptotic behavior / Diophantine approximation
Research Abstract

In this project I analyzed the spectrum of the Schrodinger operator with periodicδ'-interaction of the form
H=-d^2/dx^<2+>Σ (_1∈z) (βδ' (x-κ-2π1) 十γδ' (x-2π1) ) in L^2 R
Here, κ∈ (0,2π) and β,γ∈R-{0} are parameters, and δ' is the derivative of the Dirac delta function supported at the origin. By the periodicity of the potential of H and the Floquet-Bloch theory, the spectrum of H, denoted by σ (H), has the band structure. Let G stand for the jth gap of σ (H). We put -2τ=2π-κandκ_0 =τ/κ. The main result of this research gives a relationship between the asymptotic behavior of the length of and the number-theoretical properties of the parameter κ_0. In order to see that briefly, we introduce a number-theoretical object. Suppose that κ_0 is irrational. Let M (κ_0) stand for the Markov constant of κ_0 :
M (κ_0) =SuP{m>0 ; there exist infinitely many pairs (q,p)∈Z×N such that q|qκ_0-p|<1/ml.
This constant represents the approximability of κ_0 by rational numbers. The following implication illustrates the aforementioned relationship.
Theorem. Ifβ+γ=0,then lim inf (_j→∞)|G_1|=2π^2 (κτM (κ_0))^<-1>.

Report

(3 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • Research Products

    (24 results)

All 2008 2007 2006

All Journal Article (4 results) (of which Peer Reviewed: 2 results) Presentation (20 results)

  • [Journal Article] Spectral gaps of the Schrodinger operators with periodic δ-interactions and Diophantine approximations2007

    • Author(s)
      Kazushi Yoshitomi
    • Journal Title

      Mathematical Proceedings of the Cambridge Philosophical Society 143

      Pages: 185-199

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Spectral gaps of the SchrOdinger operators with periodic σ'-interactions and Diophantine approximations2007

    • Author(s)
      Kazushi, Yoshitomi
    • Journal Title

      Mathematical Proceedings of the Cambridge Philosophical Society 143

      Pages: 185-199

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Spectral gaps of the Schrodinger operators with periodic δ'-interactions and Diophantine approximations2007

    • Author(s)
      Kazushi Yoshitomi
    • Journal Title

      Mathematical Proceedings of the Cambridge Philosophical Society 143

      Pages: 185-199

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Spectral gaps of the Schrodinger operators with periodic δ,-interactions and Diophantine approximations2007

    • Author(s)
      KazushiYoshitom
    • Journal Title

      Mathematical Proceedings of the Cambridge Philosophical Society (掲載受理済み)

    • Related Report
      2006 Annual Research Report
  • [Presentation] Dirac operators with periodic δ-interactions-spectral gaps and inhomogeneous Diophantine approximation2008

    • Author(s)
      吉冨 和志
    • Organizer
      北東数学解析セミナー
    • Place of Presentation
      北海道大学理学部
    • Year and Date
      2008-02-21
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Dirac operators with periodic 6 -interactions -spectral gaps and inhomogeneous Diophantine approximation2008

    • Author(s)
      Gakushuin University, Mejiro, Tokyo, Japan
    • Organizer
      the 9th northeastern symposium on mathematical analysis
    • Place of Presentation
      Hokkaido University, Sapporo, Japan
    • Year and Date
      2008-02-21
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Dirac operators with periodic δ-interactions-spectral gaps and inhomogeneous Diophantine approximation2007

    • Author(s)
      吉冨 和志
    • Organizer
      学習院大学スペクトル理論セミナー
    • Place of Presentation
      学習院大学理学部
    • Year and Date
      2007-11-17
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Dirac operators with periodic σ -interactions -spectral gaps and inhomogeneous Diophantine approximation2007

    • Author(s)
      Kazushi, Yoshitomi
    • Organizer
      Seminar on spectral theory in Gakushuin University
    • Place of Presentation
      Gakushuin University, Mejiro, Tokyo, Japan
    • Year and Date
      2007-11-17
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Dirac operators with periodic δ-interactions-spectral gaps and inhomogeneous Diophantine approximation2007

    • Author(s)
      吉冨 和志
    • Organizer
      愛媛大学解析セミナー
    • Place of Presentation
      愛媛大学理学部
    • Year and Date
      2007-10-26
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Dirac operators with periodic 6 -interactions -spectral gaps and inhomogeneous Diophantine approximation2007

    • Author(s)
      Kazushi, Yoshitomi
    • Organizer
      Analysis seminar in Ehime University
    • Place of Presentation
      Ehime University, Matsuyama, Japan
    • Year and Date
      2007-10-26
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Dirac operators with periodic δ-interactions-spectral gaps and inhomogeneous Diophantine approximation2007

    • Author(s)
      吉冨 和志
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      東北大学川内北キャンパス
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Dirac operators with periodic 8 -interactions -spectral gaps and inhomogeneous Diophantine approximation2007

    • Author(s)
      Kazushi, Yoshitomi
    • Organizer
      Meetings of Mathematical Society of Japan
    • Place of Presentation
      Tohoku University, Sendai, Japan
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Dirac operators with periodic δ-interactions-spectral gaps and inhomogeneous Diophantine approximation2007

    • Author(s)
      吉冨 和志
    • Organizer
      夏の作用素論シンポジウム
    • Place of Presentation
      青森県観光物産館アスパム
    • Year and Date
      2007-09-18
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Dirac operators with periodic 6 -interactions -spectral gaps and inhomogeneous Diophantine approximation2007

    • Author(s)
      Kazushi, Yoshitomi
    • Organizer
      Summer symposium on operator theory
    • Place of Presentation
      Aomori Prefecture Tourlist Center ASPAM, Aomori, Japan
    • Year and Date
      2007-09-18
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Dirac operators with periodic δ-interactions-spectral gaps and inhomogeneous Diophantine approximation2007

    • Author(s)
      吉冨 和志
    • Organizer
      偏微分方程式集中セミナー
    • Place of Presentation
      城崎大会議館
    • Year and Date
      2007-08-01
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Dirac operators with periodic σ -interactions -spectral gaps and inhomogeneous Diophantine approximation2007

    • Author(s)
      Kazushi, Yoshitomi
    • Organizer
      Workshop on partial differential equations
    • Place of Presentation
      Kinosakidaikaigikan, Kinosaki, Japan
    • Year and Date
      2007-08-01
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Dirac operators with periodic δ-interactions-spectral gaps and inhomogeneous Diophantine approximation2007

    • Author(s)
      吉冨 和志
    • Organizer
      首都大学東京数理解析セミナー
    • Place of Presentation
      首都大学東京大学院理工学研究科
    • Year and Date
      2007-07-19
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Dirac operators with periodic σ-interactions - spectral gaps and inhomogeneous Diophantine approximation2007

    • Author(s)
      Kazushi, Yoshitomi
    • Organizer
      Analysis seminar in Tokyo Metropolitan University
    • Place of Presentation
      Tokyo Metropolitan University
    • Year and Date
      2007-07-19
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A number-theoretical aspect of the Schrodinger operator with periodic impulse2006

    • Author(s)
      Kazushi, Yoshitomi
    • Organizer
      The fifth international conference on differential equations and dynamical systems
    • Place of Presentation
      Pan American University, Edinburg, Texas, USA
    • Year and Date
      2006-12-20
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A number-theoretical aspect of the Schrodinger operator with periodic impulse2006

    • Author(s)
      吉冨 和志
    • Organizer
      The fifth international conference on differential equations and dynamical systems
    • Place of Presentation
      Pan American University,Edinburg,Texas,USA
    • Year and Date
      2006-12-17
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 周期的な点相互作用とディオファントス近似2006

    • Author(s)
      吉冨 和志
    • Organizer
      夏の作用素論シンポジウム
    • Place of Presentation
      沖縄市青年会館
    • Year and Date
      2006-07-16
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Periodic point interactions and Diophantine approximation2006

    • Author(s)
      Kazushi, Yoshitomi
    • Organizer
      Summer symposium on operator theory
    • Place of Presentation
      Okinawaseinenkaikan, Okinawa, Japan
    • Year and Date
      2006-07-16
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A number-theoretical aspect of the Schrodinger operator with periodic impulse2006

    • Author(s)
      吉冨 和志
    • Organizer
      Conference on differential and difference equations and applications 2006
    • Place of Presentation
      Rajecke Teplice,Slovak Republic
    • Year and Date
      2006-06-27
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A number-theoretical aspect of the SchrOdinger operator with periodic impulse2006

    • Author(s)
      Kazushi, Yoshitomi
    • Organizer
      Conference on differential and difference equations and applications
    • Place of Presentation
      Rajecke Teplice, Slovak Republic
    • Year and Date
      2006-06-27
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2006-04-01   Modified: 2016-04-21  

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