Orthogonal Polynomials from Hermitian Matrices
Orthogonal Polynomials from Hermitian Matrices
复制标题
DOI：
10.1063/1.2898695
复制
发表时间：
2007-12
影响因子：
1.3
通讯作者：
S. Odake;R. Sasaki
中科院分区：
文献类型：
--
作者：
S. Odake;R. Sasaki
通过有限或无限维埃尔米特矩阵的特征值问题，提出了离散变量正交多项式的统一理论。它可以被视为精确可解薛定谔方程的矩阵版本。埃尔米特矩阵（可因式分解的哈密顿量）是对应于二阶差分方程的实对称三对角（雅可比）矩阵。通过以两种不同的方式求解特征值问题，显式地建立了特征多项式及其对偶多项式的对偶关系。通过海森堡算子精确解和形状不变性技术，明确确定各种量、两类特征值（特征值和正弦坐标）、三项递推系数、归一化测度和归一化常数等。
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 Schrodinger 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.