矩阵半圆元的奇点
Singularities of matrix semicircles
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中文总结 AI 辅助
该研究针对矩阵半圆元的奇异情形，证明了其奇点指数的不变性，完成二元矩阵半圆元的分类，并揭示谱分类与代数分类的差异，给出不同类型对应的约化方法。
中文摘要 AI 辅助
设$S=\sum_{i=1}^r A_i\otimes s_i$为矩阵半圆元，其中$A_i\in M_n(\mathbb{C})$是埃尔米特系数，$s_i$是自由标准半圆生成元。其标量谱密度$f$通过Speicher方程（矩阵Dyson方程）由完全正定协方差映射$\eta_S(X)=\sum_i A_iXA_i$决定。我们研究奇异情形：此时矩阵束$\sum_i A_i x_i$是满的但非半单的，且$f$在原点处无界，与正则情形下有界的实解析密度形成对比。我们证明了三个结果：(i) 该矩阵束在合同变换$A_i\mapsto bA_ib^{*}$（$b$可逆）下，以及更一般地在协方差映射的对称缩放变换下，原点处的主导奇点指数保持不变；(ii) 对于二元矩阵半圆元（$r=2$），我们得到完整分类：在Lancaster-Rodman标准形中，每个不可分解单元属于三类之一，当$x\to0$时，$f(x)\sim c|x|^{-(n^*-1)/(n^*+1)}$，其中常数$c$显式给出，指数仅依赖于最大若尔当块的大小（有效链长度$n^*$），与耦合方式无关；结合(i)和直和性质，这对所有满二元埃尔米特矩阵束完成了分类；(iii) 谱分类严格粗于代数分类：大小为$2m$且含非实参数$\beta$的III型单元，与两个大小为$m$且参数为$|\beta|$的II型单元的直和，具有相同的标量密度，但它们的协方差映射无法通过对称缩放变换联系，标量谱无法检测$\beta$的相位。每种类型对应不同方法：I型需将Speicher方程约化为自治离散Painlevé I（McMillan）映射，II型需在分支点处进行Lyapunov-Schmidt约化，III型需通过对角酉矩阵进行规范约化。
英文摘要
Let $S=\sum_{i=1}^r A_i\otimes s_i$ be a matrix semicircular element, with Hermitian coefficients $A_i\in M_n(\mathbb{C})$ and free standard semicircular generators $s_i$. Its scalar spectral density $f$ is governed, through Speicher's equation (a matrix Dyson equation), by the completely positive covariance map $η_S(X)=\sum_i A_iXA_i$. We treat the singular regime, where the pencil $\sum_i A_i x_i$ is full but not semisimple and $f$ is unbounded at the origin, in contrast to the bounded real-analytic density of the regular case. We prove three results. (i) The leading singularity exponent at $0$ is invariant under congruence $A_i\mapsto bA_ib^{*}$ of the pencil ($b$ invertible), and more generally under symmetric scaling of the covariance map. (ii) For binary elements ($r=2$) we obtain a complete classification: in Lancaster-Rodman canonical form every indecomposable cell is of one of three types, and $f(x)\sim c|x|^{-(n^*-1)/(n^*+1)}$ as $x\to0$ with an explicit constant $c$, where the exponent depends only on the size of the largest Jordan block (the effective chain length $n^*$) and not on the coupling. With (i) and the direct-sum behaviour, this classifies all full binary Hermitian pencils. (iii) The spectral classification is strictly coarser than the algebraic one: a Type III cell of size $2m$ with non-real $β$ and the direct sum of two Type II cells of size $m$ with parameter $|β|$ have identical scalar densities, yet their covariance maps are not symmetrically scalable; the scalar spectrum cannot detect the phase of $β$. Each type calls for a different method: a reduction of Speicher's equation to an autonomous discrete Painlevé I (McMillan) map (Type I), a Lyapunov-Schmidt reduction at the branch point (Type II), and a gauge reduction by a diagonal unitary (Type III).
