Zero distribution of complex orthogonal polynomials with respect to exponential weights

Abstract

We study the limiting zero distribution of orthogonal polynomials with respect to some particular exponential weights exp(-nV(z)) along contours in the complex plane. We are especially interested in the question under which circumstances the zeros of the orthogonal polynomials accumulate on a single analytic arc (one cut case), and in which cases they do not. In a family of cubic polynomial potentials V(z) = - iz^3/3 + iKz, we determine the precise values of K for which we have the one cut case. We also prove the one cut case for a monomial quintic V(z) = - iz^5/5 on a contour that is symmetric in the imaginary axis.

arXiv comment27 pages, 13 figures

References in corpus (4)

• Asymptotics of $L_p$-norms of Hermite polynomials and Rényi entropy of Rydberg oscillator states
• The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere
• Asymptotics of orthogonal polynomials with complex varying quartic weight: global structure, critical point behaviour and the first Painleve' equation
• Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters

Cited by in corpus (1)

• S-curves in Polynomial External Fields