\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 50, pp. 1--3.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/50\hfil Eigenvalues and prime numbers]
{Eigenvalues of Sturm-Liouville operators and prime numbers}

\author[R. Amirov, I. Adalar  \hfil EJDE-2017/50\hfilneg]
{Rauf Am\.irov, \.Ibrahim Adalar}

\address{Rauf Amirov \newline
Department of Mathematics, Faculty of Sciences, 
Cumhuriyet University, 58140 Sivas, Turkey}
\email{emirov@cumhuriyet.edu.tr}

\address{Ibrahim Adalar \newline
 Zara Ahmet \c{C}uhadaro\u{g}lu Vocational School, Cumhuriyet
University, Zara/Sivas, Turkey}
\email{iadalar@cumhuriyet.edu.tr}

\thanks{Submitted June 1, 2016. Published February 20, 2017.}
\subjclass[2010]{34B05, 34B07, 11Z05}
\keywords{Sturm-Liouville; spectrum; prime numbers} 

\begin{abstract}
 We show that there is no function $q(x)\in L_2(0,1)$ which is the potential
 of a Sturm-Liouville problem with Dirichlet boundary condition whose spectrum
 is a set depending nonlinearly on the set of prime numbers as suggested by
 Mingarelli \cite{Mingarelli}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section] 
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks



\section{Introduction}

We consider the Sturm-Liouville problem
\begin{equation}
\begin{gathered} 
-y''+q(x)y=(\pi N(\lambda))^2y \\ 
y(0)=y(1)=0,  
\end{gathered} \label{SL}
\end{equation}
with
\begin{equation}
N(\lambda)=\lambda,\quad N(\lambda)=\frac{\lambda}{\ln(\lambda)},\quad
 \hbox{or}\quad N(\lambda)=li(\lambda):=\int_0^{\lambda}\frac{dt}{\ln(t)}
\end{equation}
where $li(x)$ is defined as in \cite[p. 228]{Abramowitz}. 
A real number $\lambda$ is called an eigenvalue of \eqref{SL}
 if it has a nontrivial solution. The set
of all such eigenvalues is called the spectrum of \eqref{SL} .

The purpose of this note is to prove the following results.

\begin{theorem}\label{t:1} 
If $N(\lambda) = \lambda/\ln(\lambda)$ then there is no function
$q\in L_2[  0,1]  $ such that the spectrum of \eqref{SL} is the
set of prime numbers.
\end{theorem}

\begin{theorem}\label{t:2} 
If $N(\lambda) = li(\lambda)$ then is no function $q\in
L_2[  0,1]  $ such that the spectrum of \eqref{SL} is the set of
prime numbers.
\end{theorem}

The case $N(\lambda) = \lambda$ was asked by  Zettl \cite[p.299]{Zettl}
and answered by  Mingarelli  \cite{Mingarelli}. In turn,  Mingarelli 
\cite{Mingarelli}  asked the question answered by Theorems
\ref{t:1} and \ref{t:2}.

Our proofs are based on the asymptotic distribution of prime numbers and the
asymptotic distribution of the eigenvalues for $N(\lambda)=\lambda$. In fact,
letting $\pi(x)$ denote the number of prime number less than or equal to $x$,
by the Prime Number Theorem, see \cite{Ingham}, we have
\begin{equation}
\lim_{x\to\infty} \frac{\pi(x)}{\frac{x}{\ln x}}=1 \quad
 \text{and} \quad \lim_{x\to\infty} \frac{\pi(x)}{li(x)}=1. \label{pntheo}
\end{equation}
On the other hand for $N(\lambda)=\lambda$ we have
\begin{equation}
\pi \lambda_n=n\pi+\frac{\int_0^1 q(t)dt}{2n\pi}+O(n^{-2}),
\label{spN=Id}
\end{equation}
see \cite[(3.15), p. 81]{Chadan}.

\section{Main Results}

\begin{proof}[Proof of Theorem \ref{t:1}]
Suppose the exists $q\in L_2[0,1]$ such that the spectrum of \eqref{SL} 
is the set of prime numbers.
Let $p_n$ denote the n-th prime number. By \eqref{spN=Id}, see
\cite{Chadan,Freiling,Poschel},
\begin{equation}
\Big(  \frac{\pi p_n}{\ln(p_n)}\Big)  ^2
=n^2\pi^2+\int_0^1 q(t)dt+c_n \label{asympt}
\end{equation}
where $c_n\in l_2$,

From the results by  Dusart \cite{Dusart} we have
\begin{equation}
\pi(x)\geq\frac{x}{\ln x}(1+\frac{1}{\ln x}+\frac{1.8}{\ln^2x})
\end{equation}
for $x\geq32299$. Hence
\begin{equation}
\begin{aligned} 
\lim_{n\to\infty} \Big(\big(\pi\frac{p_n}{\ln p_n}\big)^2-n^2 \pi^2\Big)
& =\lim_{n\to\infty} \Big(\big(\pi\frac{p_n}{\ln p_n}\big)^2
-(\pi(p_n))^2\pi^2\Big)\\ 
& \leq -\lim_{n\to\infty} \frac{p_n^2}{\ln^4(p_n)} 
 = -\infty. 
\end{aligned} \label{contradt:1}
\end{equation}
Since \eqref{contradt:1} contradicts \eqref{spN=Id}, the proof 
is complete.
\end{proof}


\begin{proof}[Proof of Theorem \ref{t:2}]
The classical Littlewood theorem, see \cite{Littlewood,Ingham}, proves that 
$\pi(x)-li(x)$ changes sign infinitely often. More precisely, 
it establishes the existence of increasing sequences $\{x_n\}_n$ and 
$\{y_n\}$ converging to $+\infty$  such
that
\begin{equation}
\lim_{n\to+\infty}\pi(x_n)-li(x_n)=+\infty
\quad \hbox{and}\quad   \lim_{n\to+\infty}\pi(y_n)-li(y_n)=-\infty.
\end{equation}
It is not difficult to see that if $p_j$ denotes the largest prime number
less than or equal to $x_j$ then
\begin{equation}
\lim_{n\to+\infty}\pi(p_n)-li(p_n)=+\infty.
\label{littlw+}
\end{equation}
Similarly, if $p_j$ denotes the smallest prime number greater than or equal
to $y_j$ then
\begin{equation}
\lim_{n\to+\infty}\pi(p_n)-li(p_n)=-\infty.
\label{littlw-}
\end{equation}


Assuming that the set of prime numbers is the spectrum for 
$N(\lambda)=li(\lambda)$ from \eqref{asympt} we have
\[
\lim_{n\to\infty} ((\pi li(\lambda_n))^2-n^2\pi^2)=\int_0^1 q(t)dt,
\]
which contradicts \eqref{littlw+} and \eqref{littlw-}. 
This completes the proof.
\end{proof}


\subsection*{Acknowledgements}
The authors would like to thank Alfonso Castro and the referees for their valuable
comments and remarks which led to improvements of this article.

\begin{thebibliography}{9}                                                                                                %


\bibitem{Abramowitz} M. Abramowitz, I. A. Stegun; 
\emph{Handbook of Mathematical Functions}, Dover Publications, New York, (1972).

\bibitem{Chadan} K. Chadan, D. Colton, L. Paivarinta, W. Rundell; 
\emph{An Introduction to Inverse Scattering and Inverse Spectral Problems}, 
SIAM,Philadelphia,(1997).

\bibitem{Dusart} P. Dusart; 
\emph{Autour de la fonction qui compte le nombre
de nombres premiers}, Ph.D. thesis.Universite de Limoges, (1998).

\bibitem{Freiling} G. Freiling, V. Yurko; 
\emph{Inverse Sturm-Liouville Problems and Their Applications}, NOVA 
Science Publishers, New York, (2001).

\bibitem{Ingham} A. E. Ingham; 
\emph{The distribution of prime numbers}, Cambridge Mathematical Librairy, 
Cambridge University Press, Cambridge,
(1990). Reprint of the 1932 original, With a foreword by R. C. Vaughan.

\bibitem{Littlewood} J. E. Littlewood; 
\emph{Sur la distribution des nombres premiers}, Comptes Rendus 158 (1914),
1869--1872.

\bibitem{Mingarelli} A. B. Mingarelli; 
\emph{A note on Sturm-Liouville problems whose spectrum is the set of prime 
numbers,} Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 123, pp. 1-4.

\bibitem{Poschel} J. P\"{o}schel, E. Trubowitz;
 \emph{Inverse Spectral Theory}, Academic Press, New York, (1987).

\bibitem{Zettl} A. Zettl; 
\emph{Sturm-Liouville Theory, Mathematical Surveys and Monographs},
\textbf{121}, American Mathematical Society, Rhode Island, (2005).

\end{thebibliography}


\end{document}