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

Abstract

Let E = ⋃ j = 1 l [ a 2 j − 1 , a 2 j ] be the union of l disjoint intervals and set ω ( ∞ ) = ( ω 1 ( ∞ ) , … , ω l − 1 ( ∞ ) ) , where ω j ( ∞ ) is the harmonic measure of [ a 2 j − 1 , a 2 j ] at infinity. Let μ be a measure which is absolutely continuous on E , satisfying Szegő’s condition, and with at most a finite number of point measures outside E , and denote by ( P n ) and ( Q n ) the orthonormal polynomials and their associated Weyl solutions with respect to μ . We show that the recurrence coefficients have topologically the same convergence behavior as the sequence ( n ω ( ∞ ) ) n ∈ N modulo 1. As one of the consequences, there is a homeomorphism from the so-called gaps X j = 1 l − 1 ( [ a 2 j , a 2 j + 1 ] + ∪ [ a 2 j , a 2 j + 1 ] − ) on the Riemann surface for y 2 = ∏ j = 1 2 l ( x − a j ) into the set of accumulation points of the sequence of recurrence coefficients if ω 1 ( ∞ ) , … , ω l − 1 ( ∞ ) , 1 are linearly independent over the rational numbers Q . Furthermore, it is shown that the convergence behavior of the sequence of recurrence coefficients and of the sequence of zeros of the orthonormal polynomials and Weyl solutions outside the spectrum is topologically the same. These results are proved by proving corresponding statements for the accumulation points of the vector of moments of the diagonal Green’s functions.