*Mathematical Methods in the Applied Sciences*On a conjecture of Davies and Levitin

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

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