The Stieltjes--Fekete problem and degenerate orthogonal polynomials

TL;DR

This paper generalizes Stieltjes' classical result by linking solutions of the Stieltjes--Fekete problem, involving energy minimization with logarithmic interactions, to zeros of degenerate orthogonal polynomials satisfying extra orthogonality conditions.

Contribution

It introduces a new class of degenerate orthogonal polynomials and establishes a one-to-one correspondence with solutions of the generalized Stieltjes--Fekete problem.

Findings

Solutions correspond to zeros of degenerate orthogonal polynomials

Generalization to external fields with rational derivatives

Extension of classical Stieltjes' results

Abstract

A famous result of Stieltjes relates the zeroes of the classical orthogonal polynomials with the configurations of points on the line that minimize a suitable energy. The energy has logarithmic interactions and an external field whose exponential is related to the weight of the classical orthogonal polynomials. The optimal configuration satisfies an algebraic set of equations: we call this set of algebraic equations the Stieltjes--Fekete problem or equivalently the Stieltjes--Bethe equations. In this work we consider the Stieltjes-Fekete problem when the derivative of the external field is an arbitrary rational complex function. We show that its solutions are in one-to-one correspondence with the zeroes of certain non-hermitean orthogonal polynomials that satisfy an excess of orthogonality conditions and are thus termed "degenerate". This generalizes the original result of Stieltjes.