From Schrödinger Spectra to Orthogonal Polynomials Via a Functional Equation

Abstract

The main difference between certain spectral problems for linear Schrödinger operators, e.g. the almost Mathieu equation, and three-term recurrence relations for orthogonal polynomials is that in the former the index ranges across Z and in the latter only across Z +. We present a technique that, by a mixture of Dirichlet and Taylor expansions, translates the almost Math-ieu equation and its generalizations to three term recurrence relations. This opens up the possibility of exploiting the full power of the theory of orthogonal polynomials in the analysis of Schrödinger spectra. Aforementioned three-term recurrence relations share the property that their coefficients are almost periodic. We generalize a method of proof, due originally to Jeff Geronimo and Walter van Assche, to investigate essential support of the Borel measure of associated orthogonal polynomials, thereby deriving information on the underlying absolutely continuous spectra of Schrödinger operators.