Coherent states associated with tridiagonal Hamiltonians

TL;DR

This paper explores coherent states linked to tridiagonal Hamiltonians, revealing their structure, basis expansions, and applications to systems with various spectral types, advancing the understanding of algebraic and spectral properties in quantum systems.

Contribution

It introduces a formalism for defining and analyzing coherent states associated with tridiagonal Hamiltonians, including explicit basis expansions and interpolation schemes, with applications to different spectral types.

Findings

Explicit expansion coefficients for coherent states are derived.

A complete set of special coherent states forms a basis.

Applications to systems with discrete, continuous, or mixed spectra.

Abstract

It has been shown that a positive semi-definite Hamiltonian H, that has a tridiagonal matrix representation in a given basis, can be represented in the form H = A{\dag}A, where A is a forward shift operator playing the role of an annihilation operator. Such representation endows H with rich supersymmetric properties yielding results analogous to those obtained by studying the Hamiltonian as a differential operator. Here, we study the coherent states which we define as being the eigenstates of the operator A. We explicitly find the expansion coefficients of these states in the given basis. We further identify a complete set of special coherent states which themselves can be used as basis. In terms of these special coherent states, we show that a general coherent state has the expansion form of a Lagrange interpolation scheme. As application of the developed formalism, we work out examples of systems having pure discrete, pure continuous, or mixed energy spectrum.