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Temperature dependence of the solution to the BCS gap equation for superconductivity

Research Project

Project/Area Number 24540112
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionGunma University

Principal Investigator

WATANABE Shuji  群馬大学, 大学院理工学府, 教授 (90222405)

Co-Investigator(Kenkyū-buntansha) SAITOH Saburou  群馬大学, 名誉教授 (10110397)
Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2014: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2013: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords超伝導 / BCSギャップ方程式 / 温度依存性 / 不動点定理 / 非線形積分方程式 / 超伝導のBCSモデル / 解の温度依存性 / リプシッツ連続 / 単調減少 / 温度 / 連続性
Outline of Final Research Achievements

The BCS theory is the successful quantum-mechanical theory of superconductivity. The basis of the BCS theory is the BCS gap equation.For temperatures small enough, we have shown that the solution to the BCS gap equation is continuous with respect to both the temperature and the wave vector on the basis of the Banach fixed-point theorem. For temperatures up to half of the transition temperature, we have shown that the solution is continuous with respect to both the temperature and the wave vector on the basis of the Schauder fixed-point theorem. Moreover, we have shown that the solution is Lipschitz continuous and monotonically decreasing with respect to the temperature.

Report

(4 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (18 results)

All 2014 2013 2012 Other

All Journal Article (7 results) (of which Peer Reviewed: 7 results,  Open Access: 1 results,  Acknowledgement Compliant: 3 results) Presentation (6 results) (of which Invited: 3 results) Remarks (5 results)

  • [Journal Article] Hardy inequalities for functions vanishing at negligible sets and its applications2014

    • Author(s)
      Motohiro Sobajima and Shuji Watanabe
    • Journal Title

      Applicable Analysis

      Volume: 印刷中 Issue: 1 Pages: 203-208

    • DOI

      10.1080/00036811.2014.1001381

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] An operator-theoretical treatment of the Maskawa-Nakajima equation in the massless abelian gluon model2014

    • Author(s)
      Shuji Watanabe
    • Journal Title

      Journal of Mathematical Analysis and Applications

      Volume: 418 Issue: 2 Pages: 874-883

    • DOI

      10.1016/j.jmaa.2014.04.029

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Fixed point theorems and the BCS gap equation for superconductivity2014

    • Author(s)
      Shuji Watanabe
    • Journal Title

      RIMS Kokyuroku

      Volume: 1921 Pages: 84-90

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Applications of fixed point theorems to the BCS gap equation for superconductivity2014

    • Author(s)
      S. Watanabe
    • Journal Title

      Banach and Function Spaces

      Volume: IV Pages: 425-433

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Addendum to `The solution to the BCS gap equation and the second-order phase transition in superconductivity'2013

    • Author(s)
      S. Watanabe
    • Journal Title

      Journal of Mathematical Analysis and Applications

      Volume: 405 Issue: 2 Pages: 742-745

    • DOI

      10.1016/j.jmaa.2013.03.070

    • Related Report
      2013 Research-status Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] The solution to the BCS gap equation for superconductivity and its temperature dependence2013

    • Author(s)
      S. Watanabe
    • Journal Title

      Abstract and Applied Analysis

      Volume: 2013 Pages: 1-5

    • DOI

      10.1155/2013/932085

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Temperature dependence of the solution to the BCS gap equation2012

    • Author(s)
      Shuji Watanabe
    • Journal Title

      RIMS Kokyuroku 1805, Applications of the Renormalization Group Methods in Mathematical Sciences

      Volume: RIMS Kokyuroku 1805 Pages: 1-9

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] A generalized Fourier transform and the Schroedinger operator with the inverse square potential2013

    • Author(s)
      渡辺秀司
    • Organizer
      東京理科大学神楽坂解析セミナー
    • Place of Presentation
      東京理科大学神楽坂キャンパス
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Temperature dependence of the solution to the BCS gap equation for superconductivity and fixed point theorems2013

    • Author(s)
      S. Watanabe
    • Organizer
      QMath12 (Mathematical Results in Quantum Mechanics)
    • Place of Presentation
      Humboldt University,Berlin, Germany
    • Related Report
      2013 Research-status Report
  • [Presentation] Fixed point theorems and the BCS gap equation for superconductivity2013

    • Author(s)
      S. Watanabe
    • Organizer
      RIMS Workshop ``Mathematical Aspects of Quantum Fields and Related Topics''
    • Place of Presentation
      Kyoto University, Kyoto, Japan
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Temperature dependence of the solution to the BCS gap equation for superconductivity

    • Author(s)
      Shuji Watanabe
    • Organizer
      XVII th International Congress on Mathematical Physics
    • Place of Presentation
      Aalborg, Denmark
    • Related Report
      2012 Research-status Report
  • [Presentation] Applications of fixed point theorems to the BCS gap equation for superconductivity

    • Author(s)
      Shuji Watanabe
    • Organizer
      the Fourth International Symposium on Banach and Function Spaces 2012
    • Place of Presentation
      Kyushu Institute of Technology, Kitakyushu
    • Related Report
      2012 Research-status Report
  • [Presentation] カイラル対称性の自発的破れへの不動点定理の応用

    • Author(s)
      渡辺秀司
    • Organizer
      千葉大学素粒子論研究室コロキウム
    • Place of Presentation
      千葉大学, 千葉市
    • Related Report
      2012 Research-status Report
    • Invited
  • [Remarks] 群馬大学 大学情報データベース

    • URL

      https://univ-db.media.gunma-u.ac.jp/public/main.php?pid=paper_list&cat=paper&rid=f62b69067f10dd764c52ebbaf0b7e551

    • Related Report
      2014 Annual Research Report
  • [Remarks] Cornell University Library, arXiv

    • URL

      http://arxiv.org/find/all/1/all:+AND+Shuji+Watanabe/0/1/0/all/0/1

    • Related Report
      2014 Annual Research Report
  • [Remarks] 群馬大学 大学情報データベース

    • URL

      https://univ-db.media.gunma-u.ac.jp/public/main.php?gakubu=34000&gakka=34500&kouza=&q=&btnS=%E6%A4%9C%E7%B4%A2&pid=res_search_list

    • Related Report
      2013 Research-status Report
  • [Remarks] Cornell University Library, arXiv

    • URL

      http://arxiv.org/find/all/1/all:+AND+Shuji+Watanabe/0/1/0/all/0/1

    • Related Report
      2013 Research-status Report
  • [Remarks] 群馬大学ホームページ

    • URL

      https://univ-db.media.gunma-u.ac.jp/public/main.php?pid=paper_list&cat=paper&rid=f62b69067f10dd764c52ebbaf0b7e551

    • Related Report
      2012 Research-status Report

URL: 

Published: 2013-05-31   Modified: 2019-07-29  

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