From Schrödinger spectra to orthogonal polynomials, via a functional equation

Details From Schrödinger spectra to orthogonal polynomials, via a functional equation From Schrödinger spectra to orthogonal polynomials, via a functional equation

1993-11-17

arxiv.org/abs/math/9311210v1

Mathematics

Classical Analysis and ODEs

Scientific paper

The main difference between certain spectral problems for linear Schr\"odinger operators, e.g. the almost Mathieu equation, and three-term recurrence relations for orthogonal polynomials is that in the former the index ranges across $\ZZ$ and in the latter only across $\Zp$. 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 Schr\"odinger 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 Schr\"odinger operators.

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