On a conjecture of Davies and Levitin
Hasen Mekki Öztürk · Mathematical Methods in the Applied Sciences · 2022
Let be a symmetric tridiagonal matrix with diagonal elements and off‐diagonal elements one, and be a diagonal matrix with the first diagonal elements being plus ones and the last being minus ones. Davies and Levitin studied the eigenvalues of a linear pencil as approaches to infinity. It was conjectured by DL that for any the non‐real eigenvalues of satisfy both and . The conjecture has been verified numerically for a wide range of and , but so far the full proof is missing. The purpose of the paper is to support this conjecture with a partial proof and several numerical experiments which allow to get some insight in the behaviour of the non‐real eigenvalues of . We provide a proof of the conjecture for , and also in the case where . In addition, numerics indicate that some phenomena may occur for more general linear pencils.