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Exactly Solvable Quantum Mechanics and New Orthogonal Polynomials

Research Project

Project/Area Number 25400395
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical physics/Fundamental condensed matter physics
Research InstitutionShinshu University

Principal Investigator

ODAKE Satoru  信州大学, 学術研究院理学系, 教授 (40252051)

Project Period (FY) 2013-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2013: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Keywords解ける量子力学模型 / 離散量子力学 / 例外・多添字直交多項式 / アスキースキームの直交多項式 / 再帰関係式 / 無反射ポテンシャル / 量子ダイログ関数 / 行列式表示 / 数理物理 / 例外直交多項式 / 多添字直交多項式 / アスキースキーム / 直交多項式 / 形状不変性 / Askey スキーム
Outline of Final Research Achievements

A new type of orthogonal polynomial, which forms a complete set in spite of missing degrees, has been energetically studied since the discovery in 2008, and the reporters have constructed this new type of orthogonal polynomial, multi-indexed orthogonal polynomial, for fundamental polynomials. For this new type of orthogonal polynomial, the three term recurrence relation, which is the characterization of ordinary orthogonal polynomial, does not hold, and certain recurrence relations with more terms were expected. We obtain them explicitly and the creation/annihilation operators are constructed by using them. This research and research to increase the list of multi-indexed orthogonal polynomials are done by using quantum mechanical formulation and many new exactly solvable quantum mechanical models are obtained.

Report

(5 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • 2013 Research-status Report
  • Research Products

    (25 results)

All 2017 2016 2015 2014 2013 Other

All Journal Article (14 results) (of which Peer Reviewed: 14 results,  Open Access: 8 results,  Acknowledgement Compliant: 8 results) Presentation (9 results) (of which Invited: 2 results) Remarks (2 results)

  • [Journal Article] New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics2017

    • Author(s)
      S.Odake
    • Journal Title

      Progress of Theoretical and Experimental Physics

      Volume: 印刷中 Issue: 5

    • DOI

      10.1093/ptep/ptx051

    • NAID

      120007100327

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Simplified Expressions of the Multi-indexed Laguerre and Jacobi Polynomials2017

    • Author(s)
      S.Odake and R.Sasaki
    • Journal Title

      SIGMA

      Volume: 13

    • DOI

      10.3842/sigma.2017.020

    • NAID

      120007100333

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Multi-indexed Meixner and Little q-Jacobi (Laguerre) Polynomials2017

    • Author(s)
      S.Odake and R.Sasaki
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 50 Issue: 16 Pages: 165204-165204

    • DOI

      10.1088/1751-8121/aa6496

    • NAID

      120007100334

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Recurrence Relations of the Multi-Indexed Orthogonal Polynomials IV : closure relations and creation/annihilation operators2016

    • Author(s)
      S.Odake
    • Journal Title

      Journal of Mathematical Physics

      Volume: 57 Issue: 11

    • DOI

      10.1063/1.4966985

    • NAID

      120007100332

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III2016

    • Author(s)
      S.Odake
    • Journal Title

      Journal of Mathematical Physics

      Volume: 57 Issue: 2

    • DOI

      10.1063/1.4941087

    • NAID

      120007100331

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm2015

    • Author(s)
      S.Odake and R.Sasaki
    • Journal Title

      Journal of Mathematical Physics

      Volume: 56 Issue: 7

    • DOI

      10.1063/1.4926351

    • NAID

      120007114409

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : II2015

    • Author(s)
      S.Odake
    • Journal Title

      Journal of Mathematical Physics

      Volume: 56 Issue: 5

    • DOI

      10.1063/1.4921230

    • NAID

      120007114388

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Reflectionless Potentials for Difference Schr\"odinger Equations2015

    • Author(s)
      S.Odake and R.Sasaki
    • Journal Title

      Journal of Physics

      Volume: A48 Issue: 11 Pages: 115204-115204

    • DOI

      10.1088/1751-8113/48/11/115204

    • NAID

      120007114407

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Casoratian Identities for the Wilson and Askey-Wilson Polynomials2014

    • Author(s)
      Satoru Odake and Ryu Sasaki
    • Journal Title

      J. Approximation Theory

      Volume: - Pages: 1-26

    • DOI

      10.1016/j.jat.2014.04.009

    • NAID

      120007114408

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Equivalences of the Multi-Indexed Orthogonal Polynomials2014

    • Author(s)
      S.Odake
    • Journal Title

      Journal of Mathematical Physics

      Volume: 55 Issue: 1

    • DOI

      10.1063/1.4859795

    • NAID

      120007114369

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Krein-Adler transformations for shape-invariant potentials and pseudo virtual states2013

    • Author(s)
      S. Odake and R. Sasaki
    • Journal Title

      J. Phys

      Volume: A46 Issue: 24 Pages: 245201-245201

    • DOI

      10.1088/1751-8113/46/24/245201

    • NAID

      120007114347

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Extensions of solvable potentials with finitely many discrete eigenstates2013

    • Author(s)
      S. Odake and R. Sasaki
    • Journal Title

      J. Phys

      Volume: A46 Issue: 23 Pages: 235205-235205

    • DOI

      10.1088/1751-8113/46/23/235205

    • NAID

      120007114348

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Recurrence Relations of the Multi-Indexed Orthogonal Polynomials2013

    • Author(s)
      S.Odake
    • Journal Title

      Journal of Mathematical Physics

      Volume: 54 Issue: 8

    • DOI

      10.1063/1.4819255

    • NAID

      120007114371

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Non-polynomial extensions of solvable potentials a la Abraham-Moses2013

    • Author(s)
      S. Odake and R. Sasaki
    • Journal Title

      J. Math. Phys

      Volume: 54 Issue: 10

    • DOI

      10.1063/1.4826475

    • NAID

      120007114370

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Presentation] 解ける離散量子力学模型とMeixner, little q-Jacobi(Laguerre)多添字直交多項式2017

    • Author(s)
      小竹悟
    • Organizer
      日本物理学会第72回年次大会
    • Place of Presentation
      大阪大学(大阪府大阪市)
    • Year and Date
      2017-03-20
    • Related Report
      2016 Annual Research Report
  • [Presentation] 解ける量子力学模型と一般化された閉関係式2016

    • Author(s)
      小竹悟
    • Organizer
      日本物理学会2016年秋季大会
    • Place of Presentation
      宮崎大学(宮崎県宮崎市)
    • Year and Date
      2016-09-21
    • Related Report
      2016 Annual Research Report
  • [Presentation] big q-Jacobi多項式と離散量子力学2016

    • Author(s)
      小竹悟
    • Organizer
      日本物理学会第71回年次大会
    • Place of Presentation
      東北学院大学(宮城県・仙台市)
    • Year and Date
      2016-03-20
    • Related Report
      2015 Research-status Report
  • [Presentation] 離散量子力学における無反射ポテンシャルの構成とソリトンポテンシャル2015

    • Author(s)
      小竹悟
    • Organizer
      日本物理学会2015年秋季大会
    • Place of Presentation
      大阪市立大学(大阪府・大阪市)
    • Year and Date
      2015-09-27
    • Related Report
      2015 Research-status Report
  • [Presentation] 解ける量子力学模型と |q|=1 のAskey-Wilson多項式2014

    • Author(s)
      小竹悟,佐々木隆
    • Organizer
      日本物理学会2014年秋季大会
    • Place of Presentation
      佐賀大学(佐賀県)
    • Year and Date
      2014-09-21
    • Related Report
      2014 Research-status Report
  • [Presentation] Some Properties of the Muti-indexed Orthogonal Polynomials2014

    • Author(s)
      S.Odake
    • Organizer
      Exceptional orthogonal polynomials and exact solutions in Mathematical Physics
    • Place of Presentation
      Segovia (Spain)
    • Year and Date
      2014-09-08
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] 解ける量子力学模型と新しい直交多項式の再帰関係式

    • Author(s)
      小竹悟
    • Organizer
      日本物理学会2013年秋季大会
    • Place of Presentation
      高知大学
    • Related Report
      2013 Research-status Report
  • [Presentation] 解ける量子力学模型のAbraham-Moses流の非多項式的拡張

    • Author(s)
      小竹悟,佐々木隆
    • Organizer
      日本物理学会第69回年次大会
    • Place of Presentation
      東海大学
    • Related Report
      2013 Research-status Report
  • [Presentation] Exactly Solvable Difference Schroedinger Equations

    • Author(s)
      S.Odake
    • Organizer
      Summer School on Integrability in Quantum and Statistical Systems
    • Place of Presentation
      National Taiwan University, Taiwan
    • Related Report
      2013 Research-status Report
    • Invited
  • [Remarks] Recent Papers

    • URL

      http://azusa.shinshu-u.ac.jp/~odake/paper.html

    • Related Report
      2016 Annual Research Report 2015 Research-status Report 2014 Research-status Report
  • [Remarks] Recent Papers

    • URL

      http://azusa.shinshu-u.ac.jp/~odake/paper.html

    • Related Report
      2013 Research-status Report

URL: 

Published: 2014-07-25   Modified: 2019-07-29  

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