%%%%		-*- Mode: LaTeX -*-
%%%% Author:    J. Robert Buchanan
%%%% Address:   Department of Mathematics
%%%%		Millersville University
%%%%            P.O. Box 1002
%%%%            Millersville, PA 17551-0302
%%%% Phone:     717-871-7305
%%%% FAX:	717-871-7948
%%%%
%%%% Email:     Robert.Buchanan@millersville.edu
%%%%
%%%% Date:      02/20/2025
%%%%
%%%% Purpose:   Existence of Eigenfunctions and Eigenvalues
%%%%

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\title{Existence of Eigenfunctions and Eigenvalues}
\subtitle{\textit{Partial Differential Equations}} % (optional)


\author % (optional, use only with lots of authors)
{J~Robert~Buchanan}
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\institute[Millersville University of Pennsylvania]
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{
  Department of Mathematics
}
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% - Keep it simple, no one is interested in your street address.

\date{ % (optional)
Fall
%Spring
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\the\year}

\subject{Talks}
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\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%\begin{frame}
%  \frametitle{Outline}
%  \tableofcontents
  % You might wish to add the option [pausesections]
%\end{frame}


% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:  

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

%\section{Introduction}

\begin{frame}\frametitle{Objectives}

In this lesson we will learn:
\begin{itemize}
\item to transform the regular Sturm-Liouville boundary value problem
  to an equivalent system of two first-order equations,
\item to establish the existence of an infinite sequence of
  eigenfunctions, and
\item to show the eigenvalues of the Sturm-Liouville boundary value
  problem form an increasing sequence which grows unbounded.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Sturm-Liouville Boundary Value Problem}

Throughout this discussion we will consider the regular
Sturm-Liouville BVP:
\begin{align*}
[ p(x) y'(x) ]' + (q(x) + \lambda r(x)) y(x)
	&= 0
\mbox{ for $a < x < b$}	\\
\alpha_{1} y(a) + \beta_{1} y'(a)
	&= 0	\\
\alpha_{2} y(b) + \beta_{2} y'(b)
	&= 0	.
\end{align*}

\end{frame}
%%

\begin{frame}\frametitle{Pr\''{u}fer Substitution (1 of 2)}

Convert the second-order ODE to a system of two first-order ODEs via
the substitutions:
\begin{align*}
X(x)
	&= y(x)	\\
Y(x)
	&= p(x) y'(x)	.
\end{align*}
This produces the first-order system:
\begin{align*}
%\label{eq:SL:changevariables}
X'(x)
	& = \frac{1}{p(x)} \, Y(x)	\\
Y'(x)
	& = -(q(x) + \lambda \, r(x)) X(x) 	.
\end{align*}
\uncover<2->{%
Now make the \textbf{Pr\"{u}fer substitution}:
\begin{align*}
%\label{eq:SL:changePrufer}
X(x)
	& = R(x) \sin \theta(x)	\\
Y(x)
	& = R(x) \cos \theta(x)	.
\end{align*}
}

\end{frame}
%%

\begin{frame}\frametitle{Pr\''{u}fer Substitution (2 of 2)}

The Sturm-Liouville ordinary differential equation is now written as
\begin{align*}
%\label{eq:SL:Reqn}
R'(x)
	&= \frac{1}{2} \left( \frac{1}{p(x)} - [q(x) + \lambda \, r(x)] \right)  R(x)
\sin (2 \theta(x))	\\
%\label{eq:SL:thetaeqn}
\theta'(x)
	&= \frac{1}{p(x)} \cos^{2} \theta(x) + [q(x) + \lambda \,
   r(x)] \sin^{2} \theta(x)	.
\end{align*}

\textbf{Remarks}:
\begin{itemize}
\item Any solution to this system of ODEs will solve the
  Sturm-Liouville ODE provided the change of variables is invertible.
\item The second equation is independent of $R$ and
is continuously differentiable with respect to $\theta$.
Therefore, for any initial value $\theta_{0} \in [0,2\pi)$, there
exits a unique solution, denoted as
$\theta_{\lambda}(x)$, such that $\theta_{\lambda}(a) = \theta_{0}$.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Properties of $\theta_{\lambda}(x)$}

\begin{lemma}
%\label{lma:SL:monox}
Let $\lambda$ be fixed.
Suppose $z \in (a, b)$ is such that $\theta_{\lambda}(z) = k \, \pi$ for some
$k \in \mathbb{Z}$, then $z$ is the unique solution to
$\theta_{\lambda}(x) = k \, \pi$ in $(a, b)$.
Furthermore $\theta_{\lambda}(x) < k \, \pi$ for $x < z$ and
$\theta_{\lambda}(x) > k \, \pi$ for $x > z$.
\end{lemma}

\textbf{Remark}: the function $\theta_{\lambda}(x)$ is monotone
increasing in an interval surrounding each multiple of $\pi$.

\end{frame}
%%

\begin{frame}\frametitle{Illustration}

\begin{center}
\resizebox{1.00\textwidth}{!}{\includegraphics{monotheta}}
\end{center}

\end{frame}
%%

\begin{frame}\frametitle{Comparison of Solutions}

\begin{theorem}
%\label{thm:SL:monotone}
Let $p_{i}(x)$ be continuously differentiable on $[a, b]$ and let
$q_{i}(x)$ be continuous on $[a, b]$ for $i = 1, 2$.
Suppose $0 < p_{2}(x) \le p_{1}(x)$ and $q_{1}(x) \le q_{2}(x)$ on $[a, b]$.
Let $y_{i}$ be a solution of
\[
\left[ p_{i}(x) y'(x) \right]' + q_{i}(x) y(x) = 0	,
\]
and let $R_{i}(x)$ and $\theta_{i}(x)$  be the the corresponding
solutions to the Pr\"{u}fer form of the ordinary differential equation.
If $\theta_{1}(a) \le \theta_{2}(a)$, then $\theta_{1}(x) \le
\theta_{2}(x)$ for all $x \in [a, b]$.
If in addition $q_{1}(x) < q_{2}(x)$ on $[a, b]$ then $\theta_{1}(x) <
\theta_{2}(x)$ on $(a, b]$.
\end{theorem}

\end{frame}
%%

\begin{frame}\frametitle{Transformed Boundary Conditions (1 of 2)}

Apply the change of variables and Pr\"{u}fer transformation to the
boundary condition at $x = a$.
\begin{align*}
\alpha_{1} y(a) + \beta_{1} y'(a)
	&= 0	\\
\alpha_{1} X(a) + \frac{\beta_{1} Y(a)}{p(a)}
	&= 0	\\
\frac{\alpha_{1} p(a)}{\sqrt{(\alpha_{1} p(a))^{2} + \beta_{1}^{2}}} X(a) +
\frac{\beta_{1}}{\sqrt{(\alpha_{1} p(a))^{2} + \beta_{1}^{2}}} Y(a)
	&= 0
\end{align*}
\uncover<2->{%
Without loss of generality, assume $\beta_{1} > 0$.
There exists a unique $A \in [0, \pi)$ such that
\[
\cos A
	= \frac{\alpha_{1} p(a)}{\sqrt{(\alpha_{1} p(a))^{2} + \beta_{1}^{2}}}
\mbox{ and }
\sin A
	= \frac{\beta_{1}}{\sqrt{(\alpha_{1} p(a))^{2} + \beta_{1}^{2}}}	.
\]
}

\end{frame}
%%

\begin{frame}\frametitle{Transformed Boundary Conditions (2 of 2)}

The boundary condition at $x = a$ can be written as
\begin{align*}
(\cos A) X(a) + (\sin A) Y(a)
	&= 0	\\
(\cos A) y(a) + (\sin A) p(a) y'(a)
	&= 0	.
\end{align*}
Similarly there exists a unique $B \in (0,\pi]$ such that 
the boundary condition at $x = b$ is equivalent to 
\[
%\label{eq:SL:derecheBC}
(\cos B) y(b) + (\sin B) p(b) y'(b)
	= 0	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Unboundedness of $\theta_{\lambda}(x)$}

\begin{lemma}
%\label{lma:SL:lambdatoplusinfinity}
Let $\phi_{\lambda}(x)$ be the unique solution to the
regular Sturm-Liouville differential equation
satisfying the initial conditions
\[
\phi_{\lambda}(a) = \sin A
\mbox{ and }
\phi_{\lambda}'(a) = -\frac{\cos A}{p(a)}	.
\]
If $\theta_{\lambda}(x)$ is the solution to the Pr\"{u}fer transformed ODE
corresponding to $\phi_{\lambda}(x)$ then for any $z \in (a, b]$,
\[
\lim_{\lambda \to \infty} \theta_{\lambda}(z)
	= \infty	.
\]
\end{lemma}

\end{frame}
%%

\begin{frame}\frametitle{Another Limit of $\theta_{\lambda}(x)$}

\begin{lemma}
%\label{lma:SL:lambdatominusinfinity}
Let $\phi_{\lambda}(x)$ be the unique solution to the
regular Sturm-Liouville differential equation
satisfying the initial conditions
\[
\phi_{\lambda}(a) = \sin A
\mbox{ and }
\phi_{\lambda}'(a) = -\frac{\cos A}{p(a)}	.
\]
If $\theta_{\lambda}(x)$ is the solution to the Pr\"{u}fer transformed ODE
corresponding to $\phi_{\lambda}(x)$ then for any $z \in (a, b]$,
\[
\lim_{\lambda \to -\infty} \theta_{\lambda}(z)
	= 0	.
\]
\end{lemma}

\end{frame}
%%

\begin{frame}\frametitle{Main Result}

\begin{theorem}
%\label{thm:SL:cardzero}
The eigenvalues and eigenfunctions of a regular Sturm-Liouville
boundary value problem have the following properties.
\begin{enumerate}[(i)]
\item The boundary value problem possesses an infinite sequence of
  eigenvalues $\{ \lambda_{n} \}_{n=1}^{\infty}$ that
can be arranged in increasing order as
\[
\lambda_{1} < \lambda_{2} < \cdots < \lambda_{n} <\cdots
\]
with $\lim_{n \to \infty} \lambda_{n} = \infty$.
\item Any eigenfunction $\phi_{n}(x)$ corresponding to eigenvalue
$\lambda_{n}$ has exactly $n-1$ zeroes in $[a, b]$.
\item The zeroes of $\phi_{n+1}(x)$ lie between the zeroes of $\phi_{n}(x)$.
\end{enumerate}
\end{theorem}

\end{frame}
%%

\begin{frame}\frametitle{Example}

Consider the boundary value problem:
\begin{align*}
X'' + \lambda X
	&= 0
\mbox{ for $0 < x < 1$}	\\
X(0) - \beta_{1} X'(0)
	&=0	\\
X(1) + \beta_{2} X'(1)
	&= 0	,
\end{align*}
where $\beta_{1}$ and $\beta_{2}$ are positive constants.
Show that the eigenfunctions and eigenvalues of this Sturm-Liouville
BVP have the properties outlined in the previous theorem.

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (1 of 6)}

\textbf{Case}: $\lambda = -k^{2} < 0$

\vspace*{0.10in}

The general solution to the ODE is $X(x) = c_{1} e^{k x} + c_{2} e^{-k x}$.
If $X(x)$ satisfies the boundary conditions then
\begin{align*}
c_{1} + c_{2} - \beta_{1} k (c_{1} - c_{2})
	&= 0	\\
c_{1} e^{k} + c_{2} e^{-k} + \beta_{2} k (c_{1} e^{k} - c_{2} e^{-k})
	&= 0	.
\end{align*}
The only solution to this system of equations is $c_{1} = c_{2} = 0$
and thus there are no nontrivial eigenfunctions when $\lambda < 0$.

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (2 of 6)}

\textbf{Case}: $\lambda = 0$

\vspace*{0.10in}

The general solution to the ODE is $X(x) = c_{1} x + c_{2}$.
If $X(x)$ satisfies the boundary conditions then
\begin{align*}
c_{2} - \beta_{1} c_{1}
	&= 0	\\
c_{1} + c_{2} + \beta_{2} c_{2}
	&= 0	.
\end{align*}
The only solution to this system of equations is $c_{1} = c_{2} = 0$
and thus there are no nontrivial eigenfunctions when $\lambda = 0$.

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (3 of 6)}

\textbf{Case}: $\lambda = k^{2} > 0$

\vspace*{0.10in}

The general solution to the ODE is $X(x) = c_{1} \cos (k x) + c_{2}
\sin (k x)$.
If $X(x)$ satisfies the boundary conditions then
\begin{align*}
c_{1} - \beta_{1} k c_{2}
	&= 0	\\
c_{1} \cos (k) + c_{2} \sin (k) + \beta_{2} k (-c_{1} \sin (k) + c_{2}
  \cos (k))
	&= 0	.
\end{align*}
The first equation implies $c_{1} = \beta_{1} k c_{2}$.
Substituting this into the second equation yields:
\[
c_{2} k (\beta_{1} + \beta_{2}) \cos (k) + c_{2} (1 - k^{2} \beta_{1}
\beta_{2}) \sin (k)
	= 0
\]
If $c_{2} = 0$ then $c_{1} = 0$ and there are no nontrivial eigenfunctions.

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (4 of 6)}

Assuming $c_{2} \ne 0$ then
\begin{align*}
k (\beta_{1} + \beta_{2}) \cos (k) + (1 - k^{2} \beta_{1}
\beta_{2}) \sin (k)
	&= 0	\\
\frac{k (\beta_{1} + \beta_{2})}{k^{2} - \beta_{1} \beta_{2}}
	&= \tan (k)
\end{align*}
provided $\displaystyle k \ne \frac{(2n-1) \pi}{2}$ for all $n \in \mathbb{N}$.
This is equivalent to $\displaystyle \beta_{1} \beta_{2} \ne \left(
  \frac{(2n-1) \pi}{2} \right)^{2}$ for all $n \in \mathbb{N}$.

\vspace*{0.10in}

Thus $\lambda = k^{2} > 0$ is an eigenvalue of the BVP if and only if
$k$ solves the equation
\[
\frac{k (\beta_{1} + \beta_{2})}{k^{2} - \beta_{1} \beta_{2}}
	= \tan (k)
\]

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (5 of 6)}

There are infinitely many positive eigenvalues.
\[
(n-1)^{2} \pi^{2} < \lambda_{n} < \left( \frac{(2n-1) \pi }{2} \right)^{2}
\mbox{ for $n \in \mathbb{N}$.}
\]
Numerically the first six eigenvalues are approximated as follows.
\begin{center}
\begin{tabular}{rr}
\hline
$n$	& $\lambda_{n} = k_{n}^{2}$	\\
\hline
$1$	& $1.707$	\\
$2$	& $13.492$	\\
$3$	& $43.357$	\\
$4$	& $92.769$	\\
$5$	& $161.881$	\\
$6$	& $250.719$	\\
\hline
\end{tabular}
\end{center}
The eigenfunction corresponding to eigenvalue $\lambda_{n}$ is
expressed as
\[
%\label{eq:SL:example01EF}
X_{n}(x)
	= \cos ( \sqrt{\lambda_{n}} x ) + \frac{1}{\sqrt{\lambda_{n}}}
        \sin ( \sqrt{\lambda_{n}} x )	.
\]

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (6 of 6)}

\begin{center}
\resizebox{1.00\textwidth}{!}{\includegraphics{example01lambda45}}
\end{center}

\end{frame}
%%

%\begin{frame}\frametitle{Homework}

%\begin{itemize}
%\item Read Section $x.y$
%\item Exercises: 
%\end{itemize}

%\end{frame}
%%

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