论文标题
无界变量的棕色度量，具有自由的半圆形虚拟部分
The Brown measure of unbounded variables with free semicircular imaginary part
论文作者
论文摘要
令$ x_0 $为无界的自我接合运算符，以使$ x_0 $的棕色量子在Haagerup和Schultz的意义上存在。另外，让$ \tildeσ_α$和$σ_β$分别为半圆形变量，分别为$α\ geq 0 $和$β> 0 $。假设$ x_0 $，$σ_α$和$ \tildeσ_β$都是自由独立的。我们计算$ x_0+\tildeσ_α+iσ_β$的棕色度量，扩展了最近的工作，该工作假设$ x_0 $是一个有界的自我接合随机变量。我们使用Driver，Hall和Kemp引入的PDE方法来计算棕色措施。 PDE的计算取决于棕色量度存在的一系列操作员的特征。在这种无界情况中，棕色度量具有与有限情况相同的结构。它与自由卷积$ x_0+σ_{α+β} $具有连接。我们还计算了$ x_0 $的示例。
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 $\tildeσ_α$ and $σ_β$ be semicircular variables with variances $α\geq 0$ and $β>0$ respectively. Suppose $x_0$, $σ_α$, and $\tildeσ_β$ are all freely independent. We compute the Brown measure of $x_0+\tildeσ_α+iσ_β$, 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 $x_0+σ_{α+β}$. We also compute the example where $x_0$ is Cauchy-distributed.
