Convergence Radii for Eigenvalues of Tri--diagonal Matrices

TL;DR

This paper investigates the convergence radii of eigenvalue series for a family of infinite tri-diagonal matrices, establishing bounds that depend on matrix parameters and eigenvalue index.

Contribution

It provides new bounds on the convergence radii of eigenvalue Taylor series for a class of infinite tri-diagonal matrices with off-diagonal entries depending on a parameter.

Findings

Convergence radius R_n is bounded above by C(α) n^{2-α} for 0 ≤ α < 11/6.

Eigenvalues are analytic functions of z near zero for the considered matrices.

The spectrum remains discrete for the family of matrices studied.

Abstract

Consider a family of infinite tri--diagonal matrices of the form $L+ zB,$ where the matrix $L$ is diagonal with entries $L_{kk}= k^2,$ and the matrix $B$ is off--diagonal, with nonzero entries $B_{k,{k+1}}=B_{{k+1},k}= k^\alpha, 0 \leq \alpha < 2.$ The spectrum of $L+ zB$ is discrete. For small $|z|$ the $n$-th eigenvalue $E_n (z), E_n (0) = n^2,$ is a well--defined analytic function. Let $R_n$ be the convergence radius of its Taylor's series about $z= 0.$ It is proved that $$ R_n \leq C(\alpha) n^{2-\alpha} \quad \text{if} 0 \leq \alpha <11/6.$$