Properties of new types of orthogonal polynomials and extensions of exactly solvable quantum mechanical systems

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03667
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 13010:Mathematical physics and fundamental theory of condensed matter physics-related
• Research Institution: Shinshu University
• Principal Investigator: Odake Satoru 信州大学, 学術研究院理学系, 教授 (40252051)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 例外・多添字直交多項式 / 解ける量子力学模型 / 離散量子力学 / 再帰関係式 / ダルブー変換 / ロンスキアン・カソラティアン / 自由振動子表示 / マルコフ連鎖

Outline of Final Research Achievements

We have constructed a new type of orthogonal polynomials, the multi-indexed orthogonal polynomials, which form a complete set in spite of missing degrees, using integrable deformations of quantum mechanical models, and constructed the case-(1) multi-indexed orthogonal polynomials for the remaining continuous Hahn polynomial. Then, instead of the three term recurrence relations that characterize the ordinary orthogonal polynomials, we obtained appropriate recurrence relations and constructed creation/annihilation operators. The Wronskian and Casoratian appear in the Darboux transformations that give integrable deformations of quantum mechanical models, and we showed the general identities for the Casoratian. Taking advantage of the knowledge of orthogonal polynomials obtained in these studies, we also constructed solvable Markov chains using the convolution of the weight functions of the orthogonal polynomials.

Free Research Field

数理物理学

Academic Significance and Societal Importance of the Research Achievements

直交多項式の理論は数学の一分野であるが，水素原子の量子力学に見られる様に物理学において重要な役割を果たしているだけでなく，工学などの様々な分野にも現れている。次数に欠落があるにも拘らず完全系を成す新しいタイプの直交多項式である多添字直交多項式が近年勢力的に調べられている。本研究では量子力学模型の可積分変形を利用して多添字直交多項式を構成し，その性質として再起関係式を見出し，生成消滅演算子を構成した。また，直交多項式を利用して解けるマルコフ連鎖の構成も行った。これらは直交多項式及びその周辺の話題に新しい知見をもたらした。