Spectral Theory of Jacobi Matrices in $l^2 (\mathbb{Z})$ and the $su(1,1)$ Lie Algebra

Abstract

The connection between orthogonal polynomials, continued fractions, difference equations, and self-adjoint Jacobi matrices acting in $l^2 (\mathbb{Z}^ + )$ and the extension of these connections to $l^2 (\mathbb{Z})$ are reviewed. This yields three different representations for the resolvent of the Jacobi matrix: an integral representation in terms of orthogonal polynomials, a representation in terms of continued fractions, and a representation in terms of the subdominant (or minimal) solution to the associated difference equation. This latter representation is given explicitly in terms of hypergeometric functions for the cases of associated Meixner, Meixner–Pollaczek, and Laguerre polynomials. It is also shown that it is precisely these cases that occur in the unitary irreducible representations of $su(1,1)$ for the resolvent of a real linear combination of the generators of the algebra.