Weighted equilibrium in a field of a uniform charge of an interval

Abstract

We study the logarithmic equilibrium problem on the interval $[-1,1]$ in the presence of an external field generated by a uniform background charge supported on the same interval. For a real parameter $\tau$, the external field is taken to be $\tau$ times the logarithmic potential of the unit Lebesgue measure, and for all values of $\tau$ we determine explicitly the unique equilibrium measure $\mu_\tau$, its support, its Cauchy transform, its logarithmic potential (when a closed expression is available), and the equilibrium constant. We show that the model exhibits three distinct regimes separated by critical values of $\tau$. For sufficiently negative $\tau$, the equilibrium support is a single symmetric subinterval strictly contained in $[-1,1]$. For an intermediate range of parameters, the support coincides with the full interval, and the equilibrium measure is an explicit linear combination of the Robin distribution and the Lebesgue measure. For large positive $\tau$, the support becomes disconnected and consists of two symmetric outer intervals. In each regime, we find the equilibrium measure, its Cauchy transform, its potential (when a closed expression is available), and the equilibrium constant, using complex-analytic methods and singular integral techniques. These results yield a complete picture of how the support topology and the equilibrium density/constant evolve as $\tau$ varies, including the transitions between one-cut, full-support, and two-cut configurations.