Regularity of a vector potential problem and its spectral curve

TL;DR

This paper investigates a vector measure minimization problem with external potentials, showing that equilibrium measures form a pseudo-algebraic curve with finite union of intervals, relevant to biorthogonal polynomial studies.

Contribution

It establishes the spectral curve structure of equilibrium measures in a vector potential problem with real analytic potentials.

Findings

Equilibrium measures solve a pseudo-algebraic curve.

Supports are finite unions of compact intervals.

Results apply to specific biorthogonal polynomial models.

Abstract

In this note we study a minimization problem for a vector of measures subject to a prescribed interaction matrix in the presence of external potentials. The conductors are allowed to have zero distance from each other but the external potentials satisfy a growth condition near the common points. We then specialize the setting to a specific problem on the real line which arises in the study of certain biorthogonal polynomials (studied elsewhere) and we prove that the equilibrium measures solve a pseudo-algebraic curve under the assumption that the potentials are real analytic. In particular the supports of the equilibrium measures are shown to consist of a finite union of compact intervals.