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 Mathieu 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. In the special case when they form a periodic sequence, the support can be explicitly identified by a technique due to Geronimus. The more difficult problem, when the recurrence is almost periodic but fails to be periodic, is still open and we report partial results.The main promise of the technique of this article is that it can be extended to deal with multivariate extensions of the almost Mathieu equations. However, important theoretical questions need be answered before it can be implemented to the analysis of the spectral problem for Schrödinger operators.KeywordsOrthogonal PolynomialSpectral ProblemChebyshev PolynomialBorel MeasureEssential SpectrumThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.