\documentclass[10pt]{article}

\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}





\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
            linkcolor = blue,
            urlcolor  = blue,
            citecolor = blue,
            anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
  \errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother


\usepackage{natbib}




\renewenvironment{abstract}
  {{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
  {\bigskip}

\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}


\usepackage{authblk}


\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}%

\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}

\usepackage[utf8]{inputenc}
\usepackage[english]{babel}



\usepackage{float}








\begin{document}

\title{On a conjecture of Davies and Levitin}



\author[1]{Hasen Ozturk}%
\affil[1]{Ordu University Department of Mathematics}%


\vspace{-1em}



  
  \date{May 18, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
Let \$H\_c\$ be a \$(2n)\textbackslash{}times(2n)\$ symmetric
tridiagonal matrix with diagonal elements \$c \textbackslash{}in
\textbackslash{}mathbb\{R\}\$ and off-diagonal elements one, and \$S\$
be a \$(2n)\textbackslash{}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
\$\textbackslash{}mathcal\{A\}\_c=H\_c-\textbackslash{}lambda S\$ as
\$2n\$ approaches to infinity. It was conjectured by DL that for any \$n
\textbackslash{}in \textbackslash{}mathbb\{N\}\$ the non-real
eigenvalues \$\textbackslash{}lambda\$ of
\$\textbackslash{}mathcal\{A\}\_c\$ satisfy both
\$\textbar{}\textbackslash{}lambda + c\textbar{}\textless{}2\$ and
\$\textbar{}\textbackslash{}lambda - c\textbar{}\textless{}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 \$\textbackslash{}mathcal\{A\}\_c\$. We provide
a proof of the conjecture for \$n \textbackslash{}leq 3\$, and also in
the case where \$\textbar{}\textbackslash{}lambda +
c\textbar{}=\textbar{}\textbackslash{}lambda - c\textbar{}\$. In
addition, numerics indicate that some phenomena may occur for more
general linear pencils.%
\end{abstract}%



\sloppy




\selectlanguage{english}
\FloatBarrier
\end{document}

