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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 123, pp. 1--4.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2011/123\hfil Sturm-Liouville problems]
{A note on Sturm-Liouville problems whose spectrum is the set of prime
numbers}
\author[A. B. Mingarelli\hfil EJDE-2011/123\hfilneg]
{Angelo B. Mingarelli}
\address{Angelo B. Mingarelli \newline
School of Mathematics and Statistics\\
Carleton University, Ottawa, Ontario, K1S 5B6, Canada}
\email{amingare@math.carleton.ca}
\thanks{Submitted August 9, 2011. Published September 27, 2011.}
\thanks{Partially supported by a grant from NSERC Canada}
\subjclass[2000]{34B05, 34B07, 11Z05}
\keywords{Sturm-Liouville; spectrum; prime numbers}
\begin{abstract}
We show that there is no classical regular Sturm-Liouville
problem on a finite interval whose spectrum consists of
infinitely many distinct primes numbers. In particular,
this answers in the negative a question raised by Zettl in
his book, \cite{zet}. We also show that there \emph{may}
exist such a problem if the parameter dependence is nonlinear.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\section{Introduction}
In this note a Sturm-Liouville equation on a finite interval
$[a,b]$ is defined by a second-order real differential expression
of the form
\begin{equation}\label{1.1}
-(p(x)y')' + q(x)y = \lambda r(x) y
\end{equation}
where $p, q, r : [a,b] \to \mathbb{R}$ and $1/p, q, r \in L[a,b]$,
and $\lambda$ is a generally complex parameter. By a
\emph{solution} of \eqref{1.1} we will mean, as is customary,
a function $y$ defined and absolutely continuous on $[a,b] = I$
such that $(py')(x)$ is also absolutely continuous and $y(x)$
satisfies the differential equation \eqref{1.1} almost everywhere
on $I$. In this setting one can allow $p(x)$ to have infinite
values but only on a set of measure zero.
The classical regular Sturm-Liouville problem (SLP) associated
with \eqref{1.1} consists in finding those values of
$\lambda \in \mathbb{C}$ such that \eqref{1.1} has a non-trivial
solution satisfying the \emph{separated homogeneous boundary conditions}
\begin{gather}\label{1.2}
y(a)\cos \alpha - (py')(a)\sin \alpha =0,\\
\label{1.3}
y(b)\cos \beta - (py')(b)\sin \beta =0,
\end{gather}
where $\alpha \in [0,\pi)$ and $\beta \in (0,\pi]$. Of course,
this problem has a very long history dating back to Sturm's original
contributions in the 1830's. It has been developed in many
different directions but we shall not delve on this at the moment.
Still greater generality can be obtained by allowing $p(x)$
to be identically infinite on subintervals of $I$.
In this case one needs to rewrite \eqref{1.1} as a vector system
in two dimensions, e.g.,
\begin{gather}\label{1.4}
u' = - v/p, \\
\label{1.5}
v' = (\lambda r - q) u.
\end{gather}
This defines a problem of \emph{Atkinson-type}. The boundary
conditions \eqref{1.2}-\eqref{1.3} now take the form
\begin{gather}
\label{1.6}
u(a)\cos \alpha + v(a)\sin \alpha =0,\\
\label{1.7}
u(b)\cos \beta + v(b)\sin \beta =0,
\end{gather}
where $\alpha \in [0,\pi)$ and $\beta \in (0,\pi]$.
The advantage of using the formulation \eqref{1.4}-\eqref{1.5} is
that it can be used to study three-term recurrence relations as
well, see \cite[Chapter 8]{bvp} and \cite{abm} for more details.
Here, the lack of assumptions on the signs of the coefficients
allows for the most generality. However, in this very case things
can get pretty bad since there are coefficients and a
corresponding interval $I$ with the property that the Dirichlet
problem for \eqref{1.1} on $I$ has discrete spectrum filling the
whole complex plane (see \cite{atm}, \cite{abm4}). For more
information on Sturm-Liouville problems of Atkinson-type see
\cite{bvp} and \cite{zet}.
The aim of this note is to answer, in part, Problem IV raised by
Zettl in \cite[p. 299]{zet}. We restate the problem here for ease
of reference:
\begin{quote}
Problem IV: Find a SLP whose spectrum is the primes.
\end{quote}
It is then pointed out that given any \emph{finite} set of
distinct real numbers, a SLP of Atkinson-type can be found whose
spectrum is precisely that set, \cite[p. 299]{zet}.
On the basis of spectral asymptotics and the Prime Number Theorem
we show that there does not exist a regular Sturm-Liouville
problem whose spectrum consists of infinitely many prime numbers,
thus answering said Problem IV in the negative. However, we show
that there \emph{may} exist such a problem if the parameter
dependence in \eqref{1.1} (or \eqref{1.2}-\eqref{1.3}) is
\emph{nonlinear}. Nevertheless, it is not clear how to choose the
coefficients so that the primes are generated even in such
nonlinear cases.
\section{Main results and discussions}
In the sequel the set of rational primes refers to the usual
set of prime numbers in the rational number field.
For basic results about primes and their distribution we refer
the reader to Hardy and Wright \cite{hw}. The following two results
are, for the most part, independent of sign conditions on
the coefficients in either \eqref{1.1} or \eqref{1.4}-\eqref{1.5}
(see \cite{atm} for specific details).
\begin{theorem}\label{th1}
There does not exist a regular SLP \eqref{1.1} with separated
boundary conditions \eqref{1.2}-\eqref{1.3} whose spectrum is
an infinite set of distinct rational primes.
\end{theorem}
Indeed, more is true. Theorem~\ref{th1} has a counterpart
for SLP of Atkinson-type.
\begin{theorem} \label{th2}
There does not exist a regular SLP of Atkinson-type
\eqref{1.4}-\eqref{1.5} with separated boundary conditions
\eqref{1.6}-\eqref{1.7} whose spectrum is an infinite set of
distinct rational primes.
\end{theorem}
It follows that Zettl's Problem IV as stated is unsolvable
within this framework.
Now consider the Dirichlet problem associated with \eqref{1.1}
on $[0,1]$ with nonlinear dependence
\begin{equation}\label{1.8}
-y'' + q(x)y = \big(\frac{\pi \lambda}{\log \lambda}\big)^2 y.
\end{equation}
The basic asymptotic estimate for the rational primes states that
if $p_n$ denotes the $n$-th prime, then
\begin{equation}\label{2.3}
p_n \sim n\, \log n, \quad n\to \infty,
\end{equation}
the latter being a consequence of the Prime Number Theorem,
cf., [\cite{hw}, p.10 and p.346].
Choosing the simplest case (i.e., $q(x)=0$) we get that
the eigenvalues $\lambda_ n$, for $n \geq 1$, of the Dirichlet problem
on $[0,1]$ of \eqref{1.8} must be the roots of the equation
$$
\frac{\lambda}{\log \lambda} = n,
$$
from which we deduce the approximate values of $\lambda_n$,
for large $n$, namely:
$$
\lambda_n = n\log n + n\log \log n + n \log\log\log n + \dots
$$
Even so, without any specific reference to prime numbers, the
first two terms on the right of the preceding equation already
agree with Ces\`{a}ro's approximate value for the $n$-th prime
\cite{ec}; i.e.,
$$
p_n = n\log n + n\log \log n - n + n\frac{(\log\log n - 2)}{\log n}
- \dots
$$
It is thus \emph{conceivable}, but by no means obvious, that by
modifying $q(x) \in L[0,1]$ on infinitely many subintervals on
each of which $q(x)$ takes on constant values, we may come up with
an asymptotic expansion for $p_n$ for large $n$. If so, we can
hope to find a suitable $q$ such that the Dirichlet spectrum of
\eqref{1.8} agrees with the set of all rational primes. We
emphasize that the existence of such a function $q(x)$ is an open
question.
\section{Proofs}
\begin{proof}[Proof of Theorem \ref{th1}]
We assume that $p(x)$ has a finite number of turning points and
that $(r(x)/p(x))_+ $ is not a.e. equal to zero. Here, $(\cdot)_+$
denotes the positive part of the function in question. Then, by a
direct application of the Atkinson-Mingarelli Theorem
\cite[Theorem 2.3]{atm}, we get that the positive eigenvalues,
$\lambda_n^+$, of \eqref{1.1}-\eqref{1.3} satisfy
\begin{equation}
\label{3.1}
\lambda_n^+ \sim \dfrac{n^2\,\pi^2}{\int_{a}^{b}
\sqrt{\left ({r(x)}/{p(x)} \right)_+ \, dx}}.
\end{equation}
It is now clear that \eqref{2.3} and \eqref{3.1} are incompatible
if the primes (or an infinite subsequence of distinct such)
were in fact the eigenvalues of such a problem.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{th2}]
Although the Atkinson-Mingarelli Theorem does not extend
immediately to SLP of Atkinson-type one can resort to a simple
order of magnitude argument. By assumption we take it that
the spectrum is an infinite set at the outset (more on this below).
We argue as in \cite[pp.206-207]{bvp} with minor changes.
Assuming that for the moment $u(a), v(a)$ have fixed initial values,
it is a known fact that the resulting unique solution, $u(x,\lambda)$,
and indeed $v(x, \lambda)$, are analytic for all complex $\lambda$
and thus each is an entire function of $\lambda$
(see also \cite{abm2} and \cite{abm3} for extensions of these results).
Now, following \cite[pp. 206-207]{bvp}, a straightforward calculation
shows that, even for indefinite coefficients, we still have
$$
\big| \frac{d}{dx} \log \{ |\lambda| |u|^2 + |v|^2\} \big|
\leq \sqrt{|\lambda|}(|r| + |1/p|) + |q|/ \sqrt{|\lambda|}.
$$
Integrating the latter over $[a,b]$, using the integrability
assumptions on the coefficients, and exponentiating,
we obtain the estimates (see \cite[p. 206, eq. (8.2.5)]{bvp})
$$
u(x,\lambda), v(x,\lambda)
= O\{\exp({\rm const.} \sqrt{|\lambda|)}\},
$$
thus showing that, as entire functions of $\lambda$, each must
be of order not exceeding $1/2$. It follows from the theory of entire
functions that the infinite sequence of complex zeros, $\lambda_n$,
of \eqref{1.7} (that define the eigenvalues) grow at such a rate that,
for any $\varepsilon > 0$,
$$
\sum_{n=1}^\infty \frac{1}{|\lambda_n|^{\frac{1}{2} + \varepsilon}}
< \infty.
$$
However, for sufficiently small $\varepsilon > 0$,
\eqref{2.3} implies that the series of primes
(or any infinite subsequence of distinct primes) must satisfy
$$
\sum_{n=1}^\infty \frac{1}{|p_n|^{\frac{1}{2}
+ \varepsilon}} = \infty.
$$
This contradiction completes the proof.
\end{proof}
\begin{thebibliography}{00}
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Academic Press, New York, (1964), xiv, 570 pp.
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\bibitem{abm4} A. B. Mingarelli;
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Electronic J. Diff. Eqns., 2004 (2004), no. 134, 1-8.
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\emph{Sturm-Liouville Theory},
Mathematical Surveys and Monographs \textbf{121},
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\end{thebibliography}
\end{document}