AI 中文总结
本文分析由劳伦多项式a(t)=t²+t⁻¹生成的大阶四对角托普利茨矩阵的特征值，利用Widom公式和不动点迭代法推导其渐近分布，给出原点附近及远离原点处的特征值渐近公式并经数值测试验证。
AI 中文摘要
我们对由劳伦多项式a(t)=t²+t⁻¹生成的大阶数n的四对角托普利茨矩阵的特征值进行了全面分析。这些矩阵的谱在2π/3旋转下保持不变，它们包含在复平面的三个线段中，当n趋于无穷时渐近填充这些线段。我们应用Widom行列式公式，将特征方程转化为可通过不动点迭代法求解的便利形式，之后计算特征值的渐近分布。主要结果是原点附近和远离原点处特征值的渐近公式，这些结果通过n取中等值的数值测试得到验证。
英文摘要
We perform a thorough analysis of the eigenvalues of tetradiagonal Toeplitz matrices of large order $n$ generated by the Laurent polynomial $a(t)=t^2+t^{-1}$. The spectra of these matrices are invariant under $2π/3$-rotation. They are contained in three segments of the complex plane and asymptotically fill these segments as $n$ tends to infinity. We apply Widom's formula for the determinants and transform the characteristic equation into a convenient form that can be solved by the fixed point iteration method. After that, we compute the asymptotic distribution of the eigenvalues. The main results are asymptotic formulas for the eigenvalues, both close to the origin and far from the origin. The main results are verified by numerical tests for moderate values of $n$.
Comments44 pages, 14 figures
