带状Toeplitz矩阵的极限eigenvalue分布的平衡问题
An equilibrium problem for the limiting eigenvalue distribution of banded Toeplitz matrices
摘要：
本文研究了$n\times n$带状Toeplitz矩阵的极限特征值分布为$n\to \infty$ .由Schmidt-Spitzer和Hirschman的经典结果可知，特征值在复平面上的一条特殊曲线上累积，并且归一化特征值计数测度弱收敛于该曲线上的测度$n\to\infty$ .在本文中，我们刻画了极限测度的平衡问题。极限测度是唯一测度向量的一个分量，它使定义在容许测度向量上的能量泛函最小。此外，我们还证明了每个其他组件的限制措施的规范化计数措施的某些广义本征值。
英文摘要：
We study the limiting eigenvalue distribution of $n\times n$ banded Toeplitz matrices as $n\to \infty$. From classical results of Schmidt-Spitzer and Hirschman it is known that the eigenvalues accumulate on a special curve in the complex plane and the normalized eigenvalue counting measure converges weakly to a measure on this curve as $n\to\infty$. In this paper, we characterize the limiting measure in terms of an equilibrium problem. The limiting measure is one component of the unique vector of measures that minimes an energy functional defined on admissible vectors of measures. In addition, we show that each of the other components is the limiting measure of the normalized counting measure on certain generalized eigenvalues.
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