Jost Functions for Jacobi Operators with Super-exponentially Decaying Parameters

The decay of the parameters {an, bn} for a Jacobi operator J on `(N) is related to the analyticity of the Jost function u(z; J) associated with J , which is in turn related to the spectral measure dμ of J . Damanik and Simon demonstrated the equivalence between the exponential decay of these parameters and the analyticity of the Jost function on a disk whose radius is given by the rate of decay R. In this paper, these equivalences are summarized, and an additional equivalence is shown in the case when the parameters {an, bn} decay super-exponentially, so that |an − 1| + |bn| ≤ 1/n. In this case, the Jost function will be an entire function with finite growth order no greater than 2/γ.