The Support of the Equilibrium Measure for a Class of External Fields on a Finite Interval

(Joint work with S. B. Damelin and A. B. J. Kuijlaars.)

Abstract

We investigate the support of the equilibrium measure associated with a class of nonconvex, nonsmooth (i.e. non real analytic external fields on a finite interval. Such equilibrium measures play an important role in various branches of analysis. In this paper we obtain a sufficient condition which ensures that the support consists of at most two intervals. This is applied to external fields of the form $-c \,{\sign}(x) |x|^{\alpha}$ with $c >    0$,  $\alpha \geq 1$ and $x \in [-1,1]$. If $\alpha$ is an odd integer, these external fields are smooth, and for this case the support was studied before by Deift, Kriecherbauer and McLaughlin, and by Damelin and Kuijlaars.