Convergence Radii for Eigenvalues of Tri-Diagonal Matrices

Consider a family of infinite tri-diagonal matrices of the form L + zB, where the matrix L is diagonal with entries Lkk = k2, and the matrix B is off-diagonal, with nonzero entries Bk,k+1 = Bk+1,k = kα, 0 ≤ α < 2. The spectrum of L + zB is discrete. For small |z| the nth eigenvalue En (z), En (0) = n2, is a well-defined analytic function. Let Rn be the convergence radius of its Taylor’s series about z = 0. It is proved that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_n \leq C(\alpha) n^{2-\alpha}\quad \text{if}\enspace 0 \leq \alpha <11 /6$$\end{document}.