On a conjecture of Davies and Levitin

Abstract

Let Hc$$ {H}_c $$ be a (2n)x(2n)$$ (2n)\times (2n) $$ symmetric tridiagonal matrix with diagonal elements c is an element of Double-struck capital R$$ c\in \mathbb{R} $$ and off-diagonal elements one, and S$$ S $$ be a (2n)x(2n)$$ (2n)\times (2n) $$ diagonal matrix with the first n$$ n $$ diagonal elements being plus ones and the last n$$ n $$ being minus ones. Davies and Levitin studied the eigenvalues of a linear pencil Ac=Hc-lambda S$$ {\mathcal{A}}_c={H}_c-\lambda S $$ as 2n$$ 2n $$ approaches to infinity. It was conjectured by DL that for any n is an element of N$$ n\in \mathbb{N} $$ the non-real eigenvalues lambda$$ \lambda $$ of Ac$$ {\mathcal{A}}_c $$ satisfy both |lambda+c|<2$$ \mid \lambda +c\mid and |lambda-c|<2$$ \mid \lambda -c\mid . The conjecture has been verified numerically for a wide range of n$$ n $$ and c$$ c $$, 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 Ac$$ {\mathcal{A}}_c $$. We provide a proof of the conjecture for n <= 3$$ n\le 3 $$, and also in the case where |lambda+c|=|lambda-c|$$ \mid \lambda +c\mid =\mid \lambda -c\mid $$. In addition, numerics indicate that some phenomena may occur for more general linear pencils.