2007/12/31 by Satoru Odake, Ryu Sasaki · 4 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Matrix Theory and Algorithms #Quantum Mechanics and Non-Hermitian Physics #hep-th #math-ph #math.CA #math.MP #math.QA
paper · pdf · doi:10.1063/1.2898695
published as J.Math.Phys.49:053503,2008 · 53 pages, no figures. Several sentences and a reference are added. To be published in J. Math. Phys
arxiv created 2008/02/27 · openalex publication_date 2008/05/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of Hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schrödinger equations. The Hermitian matrices (factorizable Hamiltonians) are real symmetric tridiagonal (Jacobi) matrices corresponding to second order difference equations. By solving the eigenvalue problem in two different ways, the duality relation of the eigenpolynomials and their dual polynomials is explicitly established. Through the techniques of exact Heisenberg operator solution and shape invariance, various quantities, the two types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the coefficients of the three term recurrence, the normalization measures and the normalisation constants, etc., are determined explicitly.