Orthogonal polynomials from Hermitian matrices

Journal Article · Thu May 15 04:00:00 EDT 2008 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.2898695· OSTI ID:21100285

Odake, Satoru ^[1]; Sasaki, Ryu ^[2]

---

1. Department of Physics, Shinshu University, Matsumoto 390-8621 (Japan)
2. Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502 (Japan)

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 Schroedinger 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.

• OSTI ID: 21100285
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 5 Vol. 49; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
COORDINATES
DUALITY
EIGENFUNCTIONS
EIGENVALUES
EXACT SOLUTIONS
HAMILTONIANS
HERMITIAN MATRIX
POLYNOMIALS
SCHROEDINGER EQUATION