The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element

We compute the Brown measure of $x_0+i{\sigma }_t$, where ${\sigma }_t$ is a free semicircular Brownian motion and $x_0$ is a freely independent self-adjoint element that is not a multiple of the identity. The Brown measure is supported in the closure of a certain bounded region ${\mathrm{\Omega }}_t$ in the plane. In ${\mathrm{\Omega }}_t,$ the Brown measure is absolutely continuous with respect to Lebesgue measure, with a density that is constant in the vertical direction. Our results refine and rigorize results of Janik, Nowak, Papp, Wambach, and Zahed and of Jarosz and Nowak in the physics literature. We also show that pushing forward the Brown measure of $x_0+i{\sigma }_t$ by a certain map $Q_t:{\mathrm{\Omega }}_t\to \mathbb{R}$ gives the distribution of $x_0+{\sigma }_t\mathrm{.}$ We also establish a similar result relating the Brown measure of $x_0+i{\sigma }_t$ to the Brown measure of $x_0+c_t$, where $c_t$ is the free circular Brownian motion.