Special functions and spectrum of Jacobi matrices

TL;DR

This paper explores Jacobi matrices with explicitly solvable spectra, where eigenvalues and eigenvectors are expressed via special functions, using a formalism that provides explicit characteristic functions under convergence assumptions.

Contribution

It introduces a formalism for explicitly solving spectral problems of Jacobi matrices with special functions, including new identities for q-Bessel functions.

Findings

Spectrum coincides with zeros of special functions.

Eigenvectors are expressed in terms of these special functions.

Derived identities for q-Bessel functions.

Abstract

Several examples of Jacobi matrices with an explicitly solvable spectral problem are worked out in detail. In all discussed cases the spectrum is discrete and coincides with the set of zeros of a special function. Moreover, the components of corresponding eigenvectors are expressible in terms of special functions as well. Our approach is based on a recently developed formalism providing us with explicit expressions for the characteristic function and eigenvectors of Jacobi matrices. This is done under an assumption of a simple convergence condition on matrix entries. Among the treated special functions there are regular Coulomb wave functions, confluent hypergeometric functions, q-Bessel functions and q-confluent hypergeometric functions. In addition, in the case of q-Bessel functions, we derive several useful identities.