On the Brown measure of $x+iy$, with $x,y$ selfadjoint and $y$ free Poisson

Let $x,y$ be freely independent selfadjoint elements in a $W^{\ast }$-probability space, where $y$ has free Poisson distribution of parameter $p$. We pursue a methodology for computing the absolutely continuous part of the Brown measure of $x+iy$, which relies on the matrix-valued subordination function $\mathrm{\Omega }$ of the Hermitization of $x+iy$, and on the fact that $\mathrm{\Omega }$ has an explicitly described left inverse $H$. Our main point is that the Brown measure of $x+iy$ becomes more approachable when it is reparametrized via a certain change of variable $h:\mathcal{D}\to \mathcal{M}$, with $\mathcal{D},\mathcal{M}$ open subsets of $\mathbb{C}$, where $\mathcal{D}$ and $h$ are defined in terms of the aforementioned left inverse $H$, and $\mathrm{c}\mathrm{l}(\mathcal{M})$ contains the support of the Brown measure. More precisely, we find (with some conditions on the distribution of $x$, which have to be imposed for certain values of the parameter $p$) the following formula: $$f(s+it)=\frac{1}{4\pi }[\frac{2}{t}(\frac{\mathrm{\partial }\alpha }{\mathrm{\partial }s}+\frac{\mathrm{\partial }\beta }{\mathrm{\partial }t})-\frac{2}{t}-\frac{2\beta }{t^2}],s+it\in \mathcal{M},$$ where $f$ is the density of the absolutely continuous part of the Brown measure and the functions $\alpha ,\beta :\mathcal{M}\to \mathbb{R}$ are the real and respectively the imaginary part of $h^{-1}$.