The Brown measure of the sum of a self-adjoint element and an elliptic element

Support of the Brown measure of the product of a free unitary Brownian motion by a free self-adjoint projection

The first part of this paper is devoted to the Brown measure of the product of the free unitary Brownian motion by an arbitrary free non negative operator. Our approach follows the one recently initiated by Driver-Hall-Kemp though there are substantial differences at the analytical side. In particular, the corresponding Hamiltonian system is completely solvable and the characteristic curve describing the support of the Brown measure has a non-constant (in time) argument. In the second part, we specialize our findings to the product of the free unitary Brownian motion by a free self-adjoint pro…

Brown Measures of Free Circular and Multiplicative Brownian Motions with Self-Adjoint and Unitary Initial Conditions

Let $Z_N$ be a Ginibre ensemble and let $A_N$ be a Hermitian random matrix independent from $Z_N$ such that $A_N$ converges in distribution to a self-adjoint random variable $x_0$. For each $t>0$, the random matrix $A_N+\sqrt{t}Z_N$ converges in $\ast$-distribution to $x_0+c_t$, where $c_t$ is the circular variable of variance $t$, free from $x_0$. We use the Hamilton-Jacobi method to compute the Brown measure $ρ_t$ of $x_0+c_t$. The Brown measure has a density that is constant along the vertical direction inside the support. The support of the Brown measure of $x_0+c_t$ is related to the subo…

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 $\tildeσ_α$ 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 …

两个自由随机变量之和的 Brown 测度，其中之一为三角椭圆

The Brown measure of a sum of two free random variables, one of which is triangular elliptic

本文研究了三角椭圆算子与自由随机变量之和的 Brown 测度，通过前推测度公式证明其密度绝对连续且有界，并将算子值从属函数推广至无界算子。

有限冯·诺依曼代数中椭圆算子与自由随机变量之和的布朗测度

Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra

本文研究有限冯·诺依曼代数中任意算子与自由椭圆算子之和的布朗测度，通过 Hermitian 约化和从属函数给出极限分布公式，并将结果推广到扭曲椭圆算子及 R-对角算子情形。

紧自伴线性关系的谱定理及其在椭圆边值问题中的应用

Spectral theorem for compact self-adjoint linear relations with an application to elliptic boundary value problems

该研究建立希尔伯特空间紧自伴线性关系的谱定理等相关定理，将其应用于椭圆边值问题，证明有界域Neumann拉普拉斯算子的逆为紧自伴线性关系，恢复Neumann问题解的经典本征函数展开。