ORTHOGONAL POLYNOMIALS ASSOCIATED TO ALMOST-PERIODIC SCHRODINGER-OPERATORS - A TREND TOWARDS RANDOM ORTHOGONAL POLYNOMIALS

We introduce a special class of Schrodinger type H-operators in l2 as (phi, Hpsi) = SIGMA(n=0)infinity phi(n)* [square-root R(n+1)psi(n+1) + square-root R(n)psi(n-1)], R(n) being a nonnegative real number. H satisfies the renormalization equation HD = D(H-2 - lambda), with lambda real, lambda greater-than-or-equal-to 2. D is the decimation operator defined by (phi, Dpsi) = SIGMA(n=0)infinity phi(n)*psi2n. A consequence of the renormalization equation is that the R(n) fulfil the recursion relation R0 = 0, R(2n)R2n-1 = R(n), R2n + R2n+1 = lambda. From the above relations, it can be shown that the R(n) are quasi-periodic functions of their index n. The components of the eigenfunctions of H corresponding to the eigenvalue x are the orthonormalized polynomials P(n)(x) satisfying square-root R(n+1)P(n+1)(x) + square-root R(n)P(n-1)(x) = xP(n)(x). The spectrum of H is the support of the measure associated to the polynomials. In the present case it is a compact perfect set of Lebesgue measure zero (Cantor set). It is therefore purely singular continuous. We are led to study classes of orthogonal polynomials whose three-terms recursive relations are quasi periodic functions of their index. We will present several results, conjectures and open questions which may have relevant physical applications. We study the randomness of the eigenfunctions, and we discuss their algorithmic complexity.