Zero distribution of complex orthogonal polynomials with respect to exponential weights

TL;DR

This paper investigates the zero distribution of complex orthogonal polynomials with exponential weights, identifying conditions for zeros to accumulate on a single arc and determining specific potential parameters for this behavior.

Contribution

It provides a detailed analysis of zero distributions for complex orthogonal polynomials with exponential weights, including explicit parameter conditions for the one cut case.

Findings

Identified values of K for which zeros accumulate on a single arc in cubic potentials.

Proved the one cut case for a symmetric quintic potential.

Characterized when zeros do not concentrate on a single arc.

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.