Orthogonal polynomials on several intervals: Accumulation points of recurrence coefficients and of zeros

Let E = ∪j=1[a2j−1, a2j ], a1 < a2 < ... < a2l, l ≥ 2 and set ω(∞) = (ω1(∞), ..., ωl−1(∞)), where ωj(∞) is the harmonic measure of [a2j−1, a2j ] at infinity. Let μ be a measure which is on E absolutely continuous and satisfies Szegő’s-condition and has at most a finite number of point measures outside E, and denote by (Pn) and (Qn) the orthonormal polynomials and their associated Weyl solutions with respect to dμ, satisfying the recurrence relation √ λ2+ny1+n = (x − α1+n)yn − √ λ1+ny−1+n. We show that the recurrence coefficients have topologically the same convergence behavior as the sequence (nω(∞))n∈N modulo 1; More precisely, putting (α 1+n,λ l−1 2+n) = (α[ l−1 2 ]+1+n , ..., α1+n, ..., α −[ l−2 2 ]+1+n , λ [ l−2 2 ]+2+n , ..., λ2+n, ..., λ −[ l−1 2 ]+2+n ) we prove that (α 1+nν , λ l−1 2+nν )ν∈N converges if and only if (nνω(∞))ν∈N converges modulo 1 and we give an explicit homeomorphism between the sets of accumulation points of (α 1+n,λ l−1 2+n) and (nω(∞)) modulo 1. As one of the consequences there is a homeomorphism from the so-called gaps X j=1 ([a2j , a2j+1] ∪ [a2j , a2j+1]−) on the Riemann surface y = ∏2l j=1(x−aj) into the set of accumulation points of the sequence (α 1+n,λ l−1 2+n) if the harmonic measures ω1(∞), ..., ωl−1(∞), 1 are linearly independent over the rational numbers Q. Furthermore it is demonstrated, loosely speaking, that the convergence behavior of the sequence of recurrence coefficients (α 1+n, λ l−1 2+n) and of the sequence of zeros of the orthonormal polynomials and Weyl solutions outside the spectrum is topologically the same. The above results are proved by deriving first corresponding statements for the accumulation points of the vector of moments of the diagonal Green’s functions, that is, of the sequence ( ∫ xP 2 ndμ, ..., ∫ xP 2 ndμ, √ λ2+n ∫ xP1+nPndμ, ..., √ λ2+n ∫ xP1+nPndμ)n∈N. This work was supported by the Austrian Science Fund FWF, project no. P20413-N18. The last modifications and corrections of this manuscript were done by the author in the two months preceding this passing away in November 2009. The manuscript is not published elsewhere (submitted by P. Yuditskii and I. Moale). 1 2 F. PEHERSTORFER