带状块Toeplitz矩阵特征值的平衡问题
Equilibrium problem for the eigenvalues of banded block Toeplitz matrices
摘要：
我们考虑带状块Toeplitz矩阵$T_n$与$n$块行和列。我们表明，在一定的技术假设下，规范化的本征值计数措施的$T_n$ $n\to\infty$弱收敛到一个组件的唯一向量的措施，最大限度地减少了一定的能量功能。在这种方式中，我们推广了最近的结果Duits和Kuijlaars的标量情况。一路上，我们也得到了一个平衡问题相关联的任意代数曲线，不一定是一个块Toeplitz矩阵。对于带状块Toeplitz矩阵，有几个在标量情形下不出现的新现象：（i）平衡测度的总质量不一定构成简单的等差级数，而一般是通过组合规则得到的;（ii）极限特征值分布可能含有点质量，平衡问题中可能存在吸引点源;（iii）更严重的是，有例子表明，$T_n$的极限本征值分布与平衡问题的解之间的联系中断了。我们提供了充分条件，保证没有这样的故障发生，特别是我们表明，如果$T_n$是一个海森堡矩阵。
英文摘要：
We consider banded block Toeplitz matrices $T_n$ with $n$ block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of $T_n$ for $n\to\infty$ weakly converges to one component of the unique vector of measures that minimizes a certain energy functional. In this way we generalize a recent result of Duits and Kuijlaars for the scalar case. Along the way we also obtain an equilibrium problem associated to an arbitrary algebraic curve, not necessarily related to a block Toeplitz matrix. For banded block Toeplitz matrices, there are several new phenomena that do not occur in the scalar case: (i) The total masses of the equilibrium measures do not necessarily form a simple arithmetic series but in general are obtained through a combinatorial rule; (ii) The limiting eigenvalue distribution may contain point masses, and there may be attracting point sources in the equilibrium problem; (iii) More seriously, there are examples where the connection between the limiting eigenvalue distribution of $T_n$ and the solution to the equilibrium problem breaks down. We provide sufficient conditions guaranteeing that no such breakdown occurs; in particular we show this if $T_n$ is a Hessenberg matrix.
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