论文标题
自伴侣元素和半圆形元素的虚拟倍数的棕色度量
The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element
论文作者
论文摘要
我们计算$ x_ {0}+iσ_{t} $的棕色度量，其中$σ_{t} $是一个免费的半圆布朗尼运动，$ x_ {0} $是一个自由独立的自我接球元素，不是身份的倍数。在平面中的某个有界区域$ω_{t} $的关闭中，支持棕色度量。在$ω_{t}中，$相对于lebesgue度量，棕色度量绝对连续，密度在垂直方向上是恒定的。我们的结果完善并严格化了Janik，Nowak，Papp，Wambach，Zahed以及Jarosz和Nowak在物理学文献中的结果。 我们还表明，通过某个映射$ q_ {t} $ x_ {0}+iσ_{t} $的棕色度量：ω__{t} \ rightArrow \ Mathbb {r} $提供$ x_ {0}+σ_{t}的分布。 $ x_ {0}+iσ_{t} $ to $ x_ {0}+c_ {t} $的棕色度量，其中$ c_ {t} $是免费的圆形布朗尼运动。
We compute the Brown measure of $x_{0}+iσ_{t}$, where $σ_{t}$ 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 $Ω_{t}$ in the plane. In $Ω_{t},$ 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 $x_{0}+iσ_{t}$ by a certain map $Q_{t}:Ω_{t}\rightarrow\mathbb{R}$ gives the distribution of $x_{0}+σ_{t}.$ We also establish a similar result relating the Brown measure of $x_{0}+iσ_{t}$ to the Brown measure of $x_{0}+c_{t}$, where $c_{t}$ is the free circular Brownian motion.
