Zero distribution of Bergman orthogonal polynomials for certain planar domains

Abstract

Let G be a simply connected domain in the complex plane bounded by a closed Jordan curve L and let Pn, n ≥ 0, be polynomials of respective degrees n = 0, 1,... that are orthonormal in G with respect to the area measure (the socalled Bergman polynomials). Let φ be a conformal map of G onto the unit disk. We characterize, in terms of the asymptotic behavior of the zeros of Pn's, the case when φ has a singularity on L. To investigate the opposite case we consider a special class of lens-shaped domains G that are bounded by two orthogonal circular arcs. Utilizing the theory of logarithmic potentials with external fields, we show that the limiting distribution of the zeros of the Pn's for such lens domains is supported on a Jordan arc joining the two vertices of G. We determine this arc along with the distribution function.