三对角Hermitian Toeplitz矩阵在非对角角上带扰动的特征值
Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners
AI总结：
本文研究带非对角角扰动的Hermitian Toeplitz矩阵特征值，揭示参数α对极端特征值的影响，并给出数值求解形式和渐近公式。
摘要：
本文研究首列为$2,-1,0,\ldots,0,-α$的Hermitian Toeplitz矩阵的特征值。注意到生成符号依赖于矩阵的阶数$n$。若$|α|\le 1$，则特征值属于$[0,4]$，且渐近分布为$[0,π]$上的函数$g(x)=4\sin^2(x/2)$。当$|α|>1$且$n$趋于无穷时，情况发生剧烈变化。此时两个极端特征值（最小和最大）位于$[0,4]$之外，并迅速收敛到由$α$值决定的特定极限，而其余特征值属于$[0,4]$且渐近分布为$g$。在所有情形下，我们将特征方程转化为便于数值方法求解的形式，并推导出特征值的渐近公式。
英文摘要：
In this paper we study the eigenvalues of Hermitian Toeplitz matrices with the entries $2,-1,0,\ldots,0,-α$ in the first column. Notice that the generating symbol depends on the order $n$ of the matrix. If $|α|\le 1$, then the eigenvalues belong to $[0,4]$ and are asymptotically distributed as the function $g(x)=4\sin^2(x/2)$ on $[0,π]$. The situation changes drastically when $|α|>1$ and $n$ tends to infinity. Then the two extreme eigenvalues (the minimal and the maximal one) lay out of $[0,4]$ and converge rapidly to certain limits determined by the value of $α$, whilst all others belong to $[0,4]$ and are asymptotically distributed as $g$. In all cases, we transform the characteristic equation to a form convenient to solve by numerical methods, and derive asymptotic formulas for the eigenvalues.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
