  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.
