The Brown measure of unbounded variables with free semicircular imaginary part

Let $x_0$ be an unbounded self-adjoint operator such that the Brown measure of $x_0$ exists in the sense of Haagerup and Schultz. Also let $\tilde\sigma_\alpha$ and $\sigma_\beta$ be semicircular variables with variances $\alpha\geq 0$ and $\beta>0$ respectively. Suppose $x_0$, $\sigma_\alpha$, and $\tilde\sigma_\beta$ are all freely independent. We compute the Brown measure of $x_0+\tilde\sigma_\alpha+i\sigma_\beta$, extending the recent work which assume $x_0$ is a bounded self-adjoint random variable. We use the PDE method introduced by Driver, Hall and Kemp to compute the Brown measure. The computation of the PDE relies on a charaterization of the class of operators where the Brown measure exists. The Brown measure in this unbounded case has the same structure as in the bounded case; it has connections to the free convolution $x_0+\sigma_{\alpha+\beta}$. We also compute the example where $x_0$ is Cauchy-distributed.