Superharmonic Perturbations of a Gaussian Measure, Equilibrium Measures and Orthogonal Polynomials

TL;DR

This paper studies how superharmonic perturbations of Gaussian measures influence equilibrium measures and orthogonal polynomials, revealing their support on subharmonic quadrature domains and linking zero distributions to Schwarz functions.

Contribution

It introduces a framework for analyzing superharmonic perturbations of Gaussian weights, showing equilibrium measures on quadrature domains and explicit solutions for associated orthogonal polynomial problems.

Findings

Equilibrium measures are supported on subharmonic quadrature domains.

The matrix d-bar problem for orthogonal polynomials is well-posed and explicitly solvable.

Numerical evidence links zero distributions of orthogonal polynomials to Schwarz functions.

Abstract

This work concerns superharmonic perturbations of a Gaussian measure given by a special class of positive weights in the complex plane of the form $w(z) = \exp(-|z|^2 + U^{\mu}(z))$, where $U^{\mu}(z)$ is the logarithmic potential of a compactly supported positive measure $\mu$. The equilibrium measure of the corresponding weighted energy problem is shown to be supported on subharmonic generalized quadrature domains for a large class of perturbing potentials $U^{\mu}(z)$. It is also shown that the $2\times 2$ matrix d-bar problem for orthogonal polynomials with respect to such weights is well-defined and has a unique solution given explicitly by Cauchy transforms. Numerical evidence is presented supporting a conjectured relation between the asymptotic distribution of the zeroes of the orthogonal polynomials in a semi-classical scaling limit and the Schwarz function of the curve bounding the support of the equilibrium measure, extending the previously studied case of harmonic polynomial perturbations with weights $w(z)$ supported on a compact domain.