\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 255, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/255\hfil Distribution of the Pr\"ufer angle] {Distribution of the Pr\"ufer angle in $p$-Laplacian eigenvalue problems} \author[Y.-H. Cheng, C.-K. Law, Y.-C. Luo \hfil EJDE-2014/255\hfilneg] {Yan-Hsiou Cheng, Chun-Kong Law, Yu-Chen Luo} % in alphabetical order \address{Yan-Hsiou Cheng \newline Department of Mathematics and Information Education, National Taipei University of Education, Taipei 106, Taiwan} \email{yhcheng@tea.ntue.edu.tw} \address{Chun-Kong Law \newline Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan} \email{law@math.nsysu.edu.tw} \address{Yu-Chen Luo \newline Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan} \email{leoredro@gmail.com} \thanks{Submitted November 11, 2014. Published December 4, 2014.} \subjclass[2000]{34B24, 37A30} \keywords{$p$-Laplacian eigenvalue problem; Pr\"{u}fer angle; equidistribution; \hfill\break\indent uniform distribution} \begin{abstract} The Pr\"ufer angle is an effective tool for studying Sturm-Liouville problems and $p$-Laplacian eigenvalue problems. In this article, we show that for the $p$-Laplacian eigenvalue problem, when $x$ is irrational in $(0,1)$, a sequence of modified Pr\"ufer angles (after modulo $\pi_p$) is equidistributed in $(0,\pi_p)$. As a function of $x$, $\psi_n$ is also asymptotic to the uniform distribution on $(0,\pi_p)$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \newcommand{\bfR}{\textbf{R}} \newcommand{\bfN}{\textbf{N}} \newcommand{\bfZ}{\textbf{Z}} \newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\la}{\lambda} \newcommand{\ep}{\epsilon} \newcommand{\ov}{\overline} \renewcommand{\th}{\theta} \newcommand{\ga}{\gamma} \newcommand{\de}{\delta} \newcommand{\Th}{\Theta} \newcommand{\bl}{\langle} \newcommand{\br}{\rangle} \newcommand{\Bigchi}{{\hbox{\fontsize{14pt}{12pt} \selectfont $\chi$}}} \renewcommand{\d}{} \newcommand{\rme}{{\rm e}} \section{Introduction} It is well known that when a real number $x$ is irrational, the sequence $\{ x_n=\bl nx\br\}$ is dense in $(0,1)$. Here for any $t\in \mathbb{R}$, the fractional part of $t$ is denoted by $\bl t\br:=t-[t]$. It is equivalent to saying that $\{\xi_n=\sin(n\pi x)\}$ is dense in $[-1,1]$. Furthermore, the above sequence $\{ x_n\}$ is equidistributed in $(0,1)$ in the sense below (\cite[p.105]{SS}). \noindent\textbf{Definition.} A sequence $\{{x}_n\}\subset (0,1)$ is said to be equidistributed in $(0,1)$ if for any subinterval $(a,b)\subset (0,1)$, $$ \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N \Bigchi_{(a,b)}(x_n)=b-a. $$ The above property is a basic one in ergodic theory. It tells us that the sequence spreads evenly in the interval $(0,1)$. In fact, this equidistribution theorem is equivalent to the property that for any $f\in L^1(0,1)$, $$ \int_0^1 f(x)\, dx=\lim_{N\to\infty}\frac{1}{N} \sum_{n=1}^N f(x_n), $$ which in term is equivalent to saying that the transformation $T(\th)=\langle \th+x\rangle$ is ergodic \cite{SS,BS}. Consider the Sturm-Liouville problem \begin{equation} -u''+q(x)u=\la u \label{eq1.1} \end{equation} subject to boundary conditions \begin{equation} \begin{gathered} u(0)\cos\al + u'(0)\sin\al = 0\\ u(1)\cos\be + u'(1)\sin\be = 0 \end{gathered} \label{eq1.2} \end{equation} where $\al, \be\in[0, \pi)$, and $q\in L^{1}(0,1)$. We call $\al,\be$ the boundary phases. The Pr\"ufer substitution \begin{equation} u=r(x)\sin\th(x),\quad u'=r(x)\cos\th(x)\label{eq1.25} \end{equation} is a useful method to study the Sturm-Liouville problem, such as the existence of countably many simple eigenvalues, oscillations of the $n$th {eigenfunction}, the asymptotics of the eigenvalues and eigenfunctions \cite{BR}. In \cite{a64}, Atkinson showed that the Sturm-Liouville properties are also valid when the coefficient function $q$ is $L^1$. His method is also this Pr\"ufer substitution in spirit. Furthermore, Binding and Volkmer \cite{BV12} (see also \cite{BV13}) showed that one can use the Pr\"ufer substitution method to show the distribution of periodic and anti-periodic eigenvalues for periodic Sturm-Liouville problems. (Traditionally the Hill discriminant function is used to prove this distribution.) Thus the Pr\"ufer angle is an effective tool for the Sturm-Liouville theory. It would be interesting to explore further properties of this Pr\"ufer angle. In this paper, we shall study the equidistribution property. In recent years, the Pr\"ufer angle has been used to show that another class of degenerate boundary value problems, the $p$-Laplacian eigenvalue problem, observes the Sturm-Liouville properties, as shown by Binding and Drabek \cite{BD03} (see also \cite{bd06}). Let $(\lambda_n,y_n)$ be the $n$th eigenpair of the boundary value problem \begin{equation} \begin{gathered} -(|y'|^{p-2}y')'=(p-1)(\lambda-q(x))|y|^{p-2}y,\\ y(0)S'_p(\alpha) + y'(0)S_p(\alpha) = 0,\\ y(1)S'_p(\beta) + y'(1)S_p(\beta) = 0. \end{gathered} \label{eq1.3} \end{equation} Here, $S_p$ is called the generalized sine function and defined as the solution of the initial value problem \begin{gather*} (|S_p'(x)|^{p-2}S_{p}'(x))'+(p-1)|S_p(x)|^{p-2}S_{p}(x) = 0,\\ S_p(0)=S_p'(0)-1= 0. \end{gather*} It is known that the function $S_p$ is $2\pi_p$-periodic on $\mathbb{R}$, where $$ \pi_p\equiv 2\int_0^1 (1-t^p)^{-1/p} dt, $$ and for all $x\in \mathbb{R}$, the following identity holds: $$ |S_p(x)|^p+|S_p'(x)|^p=1. $$ Note that $\pi_p$ is strictly decreasing in $p$ \cite{bd06}. When $p=2$, we have $\pi_2=\pi$ and $S_p(x)=\sin x$. Moreover, for $q=0$ and $p=2$, the Dirichlet eigenvalues and eigenfunctions are $\la_n=(n\pi)^2$ and $y_n=\sin(n\pi x)$. For $a>0$, let us define the fractional part of $ \bl t\br_a$ $(t \pmod a)$ by $$ \bl t\br_a:= t-a\cdot\left[t/a\right]. $$ When $a=1$, we denote this fractional part simply by $ \bl t\br$. As discussed above, when $x$ is irrational, the sequence $\{ \bl n\pi x\br_\pi\}$ is equidistributed in $(0,\pi)$. We shall see that a sequence of modified Pr\"ufer angles $\{\bl \psi_n(x)\br_{\pi_p}\}$ also observe this ergodic behavior. Consider the modified Pr\"ufer substitution \begin{equation} y(x)=R(x)S_p(\psi(x)),\quad y'(x)=\lambda^{1/p}R(x)S'_p(\psi(x)).\label{eq1.35} \end{equation} We call $\psi(x)$ the modified Pr\"ufer angle at $x$ of the problem \eqref{eq1.3}. It becomes $\psi_n(x)$ when associated with the $n$th eigenpair $(\la_n,y_n)$. We note that in literature $\psi(x)$ can also help to give estimates for the eigenvalues and nodal points. See \cite{CCL}. \begin{theorem} \label{thm1.1} Fix any irrational number $x\in (0,1)$. For any boundary phases $\al$ and $\be$, the sequence $ \{\bl \psi_n(x)\br_{\pi_p}\}$ is equidistributed in $(0,\pi_p)$. \end{theorem} We remark that $\psi_n(x)$ can be viewed as the phase of the eigenfunction $y_n$ at $x$, analogous to the argument of the function $\sin(n\pi x)$. Moreover $\psi_n(x)$ demonstrates another property of uniform distribution, just like $\bl n\pi x\br_\pi$. \begin{theorem} \label{thm1.2} For $q\in L^{1}(0, 1)$, the distribution of the modifed Pr\"ufer angle $\psi_{n}$ defined in \eqref{eq1.35} is asymptotic to the uniform distribution on $(0, \pi_p)$. That is, for all $t\in(0,\pi_p)$, $$ P_{n}(t):=\mu\{x\in(0, 1): \bl\psi_{n}(x)\br_{\pi_p} < t\}\to\frac{t}{\pi_p}\quad \text{as } n\to\infty\, . $$ Here $\mu$ denotes the Lebesgue measure on $\mathbb{R}$. \end{theorem} The above two theorems are the main results of this paper. To prove them, we need to use the following lemma. Define $CT_{p} (x) \equiv S'_{p}(x)/S_{p}(x)$ and let $CT^{-1}_{p} (x)$ be the inverse function of $CT_{p} (x)$ taking value in $(0,\pi_p)$. \begin{lemma} \label{lem1.3} The modified Pr\"{u}fer angle $\psi_n(x)$, defined in \eqref{eq1.35} for the $p$-Lapla\-cian eigenvalue problem \eqref{eq1.3}, has the asymptotic formula \begin{equation} \psi_n(x)=\lambda_n^{1/p}x+\psi_n(0)+O(\frac 1{\lambda_{n}^{1-1/p}}), \label{eq1.4} \end{equation} where $$ \psi_n(0)=\begin{cases} 0, & \text{if } \alpha=0,\\ CT_p^{-1}(-\frac{CT_p(\alpha)}{\lambda_n^{1/p}}),& \text{if }\al>0. \end{cases} $$ \end{lemma} \begin{proof} Since $\frac{y'(x)}{\lambda^{1/p}y(x)}=\frac{S'_p(\psi(x))}{S_p(\psi(x))}$, differentiating both sides with respect to $x$, we have \begin{equation} \psi'(x)=\lambda^{1/p}-\frac{q(x)}{\lambda^{1-1/p}}|S_p(\psi(x))|^p =\lambda^{1/p}+O(\frac 1 {\lambda^{1-1/p}}). \label{eq1.5} \end{equation} Integrating \eqref{eq1.5} with respect to the $n$th eigenfunction from $0$ to $x$ and we have \begin{equation} \psi_n(x)=\lambda_n^{1/p}x+\psi_n(0)+O(\frac 1 {\lambda_{n}^{1-1/p}}). \label{eq1.6} \end{equation} This completes the proof. \end{proof} \noindent\textbf{Remark.} If the eigenvalues $\la_n\to\infty$, then when $\al>0$, \begin{equation} \lim_{n\to\infty} \psi_n(0)=\frac{\pi_p}{2}.\label{eq1.8} \end{equation} In section 2, we shall prove Theorem \ref{thm1.2}. The proof of Theorem~\ref{thm1.1}, using Weyl's criterion, will be given in section 3. In section 4, we shall see that the classical Pr\"ufer angle {$\theta_n(x)$ after modulo $\pi_p$ is not equidistributed in $(0,\pi_p)$. Nor is the sequence asymptotic to the uniform distribution.} The question that whether the classical Pr\"ufer angle is dense in $(0,\pi_p)$ or not is still open. The problem seems to be related to continued fractions with bounded and unbounded elements. It would be interesting to study this question. As discussed above, the eigenvalues and eigenfunctions of the Sturm-Liouville operators $H_q$ behaves like $H_0$, the case when the potential function $q=0$. Say, with Dirichlet boundary conditions has the asymptotics $y_n\sim A\sin(n\pi x)$ and the nodal points $ x^{(n)}_k\sim \frac{k}{n}$. For these asymptotic results, the use of another modified Pru\"fer angle $\phi_n=\psi_n/\sqrt{\la_n}$ so that $$\phi_n'=1-\frac{q}{\la_n}\sin^2(\sqrt{\la_n}\phi_{n}(x)), $$ gives the simplest proof. The situation with the $p$-Laplacian operator is analogous. This paper establishes another analogy of equidistribution between $\bl\psi_n(x) \br_\pi$, and $\bl n\pi x\br_\pi$ which is associated with $q=0$. It supports the fact that $\psi_n/\sqrt{\la_n}$ was a better choice. \section{Proof of Theorem \ref{thm1.2}} \begin{lemma}\label{lem2.1} For any $t\in(0,\pi_p)$, $a>0$, $b\in\mathbb{R}$, we have \begin{itemize} \item[(a)] $ \mu\{x\in(0,\pi_p):\bl x+b\br_{\pi_p}0=\beta\\ n-1,&\text{if } \alpha,; \beta>0\,, \end{cases} $$ and, for any $\gamma\in[0, \pi_p)$, $$ \widetilde{CT_p}(\ga)^{(p-1)}=\begin{cases} 0 & \text{if }\gamma=0\\ |CT_p(\ga)|^{p-2}CT_p(\ga) & \text{if }\gamma>0\,. \end{cases} $$ \end{lemma} \begin{proof}[Proof of Theorem \ref{thm1.2}] From \eqref{eq1.6} and \eqref{eq2.1}, $$ \psi_n(x )= \lambda_n^{1/p}x+\psi_n(0)+o(1) =n_{\alpha\beta}\pi_{p}x+\psi_{{n}}(0)+o(1). $$ Hence by Lemma \ref{lem2.1}, \begin{align*} P_n(t)&:={\mu}\{x\in(0,1):\bl\psi_n(x)\br_{\pi_p}0$, there exists a $N_1\in\mathbb{N}$ such that for all $N\geq N_1$, \[ \big|\frac 1 N\sum^{N}_{n=1}b_n-b\big|\leq\frac\varepsilon 3. \] On the other hand, $a_n=b_n+o(1)$ as $n\to\infty$. Given $\varepsilon>0$, there exists a $N_2\in\mathbb{N}$ such that for all $n\geq N_2$, $|a_n-b_n|\leq\frac\varepsilon 3$. Now let \[ M=\sum^{N_2-1}_{n=1}|a_n-b_n|. \] Let $N_0\in\mathbb{N}$ be such that $N_{0}\geq \max\{N_1, N_2, \frac{3M}{\varepsilon}\}$. Then for all $N\geq N_0$, \begin{align*} \big|\frac 1 N\sum^N_{n=1}a_n-b\big| &\leq \big|\frac 1 N\sum^N_{n=1}(a_n-b_n)\big| +\big|\frac{1}{N}\sum^N_{n=1}b_n-b\big|\\ &< \frac{M+(N-N_2+1)\cdot\varepsilon/3}{N}+\frac{\varepsilon}{3} < \varepsilon\,. \end{align*} This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] By Theorem \ref{th3.1}, it suffices to show that $$ \lim_{N\to\infty} \sum_{n=1}^N \exp\Big(\frac{2i k\pi\psi_n(x)}{\pi_p}\Big)=0. $$ Fixed $x\in\mathbb{R}$, from \eqref{eq1.6} and \eqref{eq2.1}, $$ \psi_n(x)=\lambda_n^{1/p}x+\psi_n(0)+o(1)=n_{\alpha\beta}\pi_px+\psi_n(0)+o(1). $$ If $\al=\be=0$, then $\psi_n(0)=0$ and $n_{\al\be}=n$. Hence $$ \psi_n(x)=n\pi_p x+o(1). $$ Since $\{ \bl n\pi_p x\br_{\pi_p}\}$ is equidistributed in $(0,\pi_p)$, by Lemma \ref{lem3.2}, {$\{ \bl \psi_n(x)\br_{\pi_p}\}$} is also equidistributed. If $\al>0=\be$, then by \eqref{eq1.8}, $\psi_n(0)=\frac{\pi_p}{2}+o(1)$, and $ n_{\al\be}=n-\frac{1}{2}$. Thus $$ \psi(x)=(n-\frac{1}{2})\pi_p x+\frac{\pi_p}{2}+o(1). $$ So when $x\in (0,1)$ is irrational, by taking any $k\in\mathbb{Z}\setminus \{ 0\}$, $$ \frac{1}{N} \sum_{n=1}^N \exp\Big(2\pi i k(n-\frac{1}{2}){x}+\pi i k\Big) =\rme^{\pi ik(1-x)}\cdot \frac{1}{N}\sum_{n=1}^N\exp(2\pi ikn x), $$ which converges to $0$ as $N\to\infty$. By Weyl's criterion, $\{\bl\psi_n(x)\br\}_{\pi_p}$ is also equidistributed. The other cases $\al=0<\be$ and $\al,\be>0$ are similar. \end{proof} \section{Classical Pr\"ufer angle} The classical Pr\"ufer angle $\th(x)$ is defined through $$ y=R(x)S_p(\th(x)),\quad y'=R(x)S_p'(\th(x)), $$ and the Pr\"ufer angle $\th_{n}(x)$ associated with the $n$th eigenpair satisfies \begin{equation} CT_p(\th_n(x))= \la_n^{\frac{1}{p}} \, CT_p(\psi_n(x)). \label{eq4.10} \end{equation} We denote $$ b_n:= \bl\th_n(x)\br_{\pi_p} =CT_p^{-1} \Big( \la_n^{\frac{1}{p}} CT_p({n_{\alpha\beta}\pi_p x+\psi_n(0)+o(1)})\Big), $$ taking value of the inverse function $CT_p^{-1}$ in $(0,\pi_p)$. \begin{theorem} \label{thm4.1} For $x\in(0,1)$, the sequence of classical Pr\"ufer angle $ \{ \bl\th_n(x)\br_{\pi_p}\}$ is NOT equidistributed in $(0,\pi_p)$. Nor asymptotic to the uniform distribution. \end{theorem} \begin{proof} Let $I$ be the subinterval $ ( CT_p^{-1}(1), \pi_{p}/2)\subset (0,\pi_p/2)$. We shall see that for any $x\in (0,1)$, \begin{equation} \lim_{N\to\infty}\sum_{n=1}^N \Bigchi_I(b_n) \neq {\frac{1}{2}-\frac{CT_p^{-1}(1)}{\pi_p}}.\label{eq4.1} \end{equation} Observe that \begin{equation} \begin{aligned} \Bigchi_I(b_n)=1 &\Leftrightarrow CT_p^{-1}\left( \la_n^{1/p} CT_p(\psi_n(x)) \right)\in I = \left(CT_p^{-1}(1),\frac{\pi_{p}}{2}\right)\\ &\Leftrightarrow \lambda_n^{1/p} CT_p(\psi_n(x))\in \left(0,1\right)\\ &\Leftrightarrow \bl n_{\al\be}\pi_p x{+\psi_n(0)}+o(1)\br_{\pi_p} \in \big( CT_p^{-1}( \la_n^{-1/p}),\frac{\pi_{p}}{2} \big) \end{aligned}\label{eq4.2} \end{equation} If $\al=\be=0$, then $n_{\al\be}=n$ and $\psi_n(0)=0$. Hence $\Bigchi_I(b_n)=1$ if and only if $$ {\langle\psi_n(x)\rangle_{\pi_p}= \bl n\pi_p x+o(1)\br_{\pi_p}\in J_n:=}\left( CT_p^{-1}\left( \la_n^{-1/p} \right),\frac{\pi_{p}}{2} \right). $$ Since $ \lim_{n\to\infty}CT_p^{-1}( \la_n^{-1/p}) = \frac{\pi_p}{2}$, the probability of $b_n$ in $I$ tends to $0$ as $n\to\infty$. Therefore. $$ \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N \Bigchi_I(b_n) {=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\Bigchi_{J_n}(\langle\psi_n(x)\rangle_{\pi_p})=0< \frac{1}{2}-\frac{CT_p^{-1}(1)}{\pi_p},} $$ because $|J_n|\to 0$ as $n\to\infty$. In case $\al>0=\be$, by \eqref{eq4.2}, \begin{eqnarray*} \Bigchi_I(b_n)=1 &\Leftrightarrow \bl (n-\frac{1}{2})\pi_p x+\frac{\pi_p}{2}+o(1)\br_{\pi_p} \in \left( CT_p^{-1}\left( \la_n^{-1/p} \right),\frac{\pi_{p}}{2} \right)\\ &\Leftrightarrow \bl (n-\frac{1}{2})\pi_p x+o(1)\br_{\pi_p} \in \left(\frac{\pi_{p}}{2}+CT_p^{-1}(\la_n^{-1/p}) ,\pi_p\right), \end{eqnarray*} by Lemma \ref{lem2.1}(a). Therefore by a similar argument as above, $$ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N \Bigchi_I(b_n)=0<{\frac{1}{2}- \frac{CT_p^{-1}(1)}{\pi_p}} . $$ Therefore, \eqref{eq4.1} is also valid. The other cases are similar. On the other hand, from \eqref{eq4.10}, \begin{align*} \bl\th_n(x)\br_{\pi_p}