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\section{Sturm-Liouville Theory}

\subsection{Review of second-order linear ODEs}

We wish to solve general inhomogeneous ODE
\[ \mathcal{L} y \equiv \alpha(x) y'' + \beta(x)
y' + \gamma(y) y = f(x) \tag{2.1} \]
\begin{itemize}
    \item The \emph{homogeneous equation}
        \[ \boxed{\mathcal{L} y \equiv 0 }
        \tag{2.2} \]
        has two independent solutions $y_1(x)$,
        $y_2(x)$ (besides trivial $y \equiv 0$),
        with the \emph{complementary function}
        $y_c(x)$ the general solution of (2.2):
        \[ y_c(x) = A y_1(x) + B y_2(x) \tag{2.3} \]
        where $A, B$ are arbitrary constants.
    \item The inhomogeneous equation
        \[ \mathcal{L} y = f(x) \tag{2.4} \]
        (i.e. the driving force or source term
        $f(x)$) has a special solution called the
        particular integral $y_p$. The general
        solution of (2.4) is then
        \[ y(x) = y_p(x) + A y_1(x) + B y_2(x)
        \tag{2.5} \]
    \item Two \emph{boundary} or initial data are
        required to determine $A$, $B$.
        \begin{enumerate}[(a)]
            \item \emph{Boundary conditions} (BC)
                Solve (2.4) on $a < x < b$ given
                $y$ at $x = a, b$ (Dirichlet) \emph{or}
                specify $y'$ at $x = a, b$
                (Neumann) \emph{or} mixed $y +
                ky'$ etc. Homogeneous boundary
                conditions are often assumed,
                $y(a) = y(b) = 0$ to admit the
                trivial solution $y \equiv 0$. Can
                be achieved by adding
                complementary function (2.3).
                \[ \tilde{y} = y + A y_1 + B y_2 \]
                such that $\tilde{y}(a) =
                \tilde{y}(b) = 0$.
            \item Alternatively we may be given
                \emph{initial conditions} Solve
                (2.4) for $x \ge a$, given $y, y'$
                at $x = a$.
        \end{enumerate}
\end{itemize}

\subsubsection*{General eigenvalue problem}

To solve (2.1) employing eigenfunction expansions
(like Fourier series (1.22)) we must solve the
\emph{related} eigenvalue problem
\[ \alpha(x) y'' + \beta(x)y' + \gamma(x)y =
-\lambda \rho(x) y \tag{2.6} \]
with specified boundary conditions. This form
often occurs in higher dimensions after seperation
of variables.

\subsection{Self-adjoint operators}

\begin{definition*}[Inner product]
    For two (complex-valued) functions $f, g$ on
    $a \le x \le b$ define
    \[  \langle f, g \rangle = \int_a^b f^*(x)
    g(x) \dd x \]
    (later $f, g$ assumed to be real so we will
    drop the complex conjugate part). The norm of
    $f$ is $\|f\| = \sqrt{\langle f, f \rangle}$.
\end{definition*}

\subsubsection*{The Sturm-Liouville equation}

The eigenvalue problem (2.6) greatly simplifies if
$\mathcal{L}$ is self-adjoint, i.e. it can be
expressed in \emph{Sturm-Liouville form}
\begin{flashcard}[sturm-liouville-form]
\prompt{Sturm Liouville form?}
\[ \boxed{\mathcal{L} y \equiv \cloze{-(py')' + qy =
\lambda \omega y} \tag{2.7}} \]
\cloze{where $\omega(x)$ is \fcemph{non-negative}.}
\end{flashcard}
where the \emph{weight function} $\omega(x)$ is
non-negative $\omega(x) \ge 0$ for all $x$.
\myskip
Converting to Sturm-Liouville form: Multiply (2.6)
by an integrating factor $F(x)$ to find
\[ F\alpha y'' + F\beta y' + F\gamma y = -\lambda
F \rho y \]
\[ \dfrac{}{x} (F\alpha y') - F'\alpha y' -
F\alpha'y' + F\beta y' + F\gamma y = -\lambda
F\rho y \]
We want to eliminate the $y'$ term, so we want
\[ F'\alpha = F(\beta - \alpha') \implies
\frac{F'}{F} = \frac{\beta - \alpha'}{\alpha} \]
so
\begin{hiddenflashcard}[coefficients-for-converting-to-SL]
\prompt{What are $\alpha, \beta, \gamma, \lambda,
\rho$ for converting to SL form?}
\[ \cloze{\alpha y'' + \beta y' + \gamma y =
-\lambda \rho y} \]
\end{hiddenflashcard}
\begin{hiddenflashcard}[p-converted-to-SL]
\prompt{What is $p$ after converting to SL form?}
\[ p = \cloze{F\alpha} \]
\end{hiddenflashcard}
\begin{hiddenflashcard}[q-converted-to-SL]
\prompt{What is $q$ after converting to SL form?}
\[ q = \cloze{-F\gamma} \]
\end{hiddenflashcard}
\begin{hiddenflashcard}[omega-converted-to-SL]
\prompt{What is $\omega$ after converting to SL
form?}
\[ \omega = \cloze{F\rho} \]
\end{hiddenflashcard}
\begin{flashcard}[integrating-factor-to-SL]
\prompt{Integrating factor to SL form?}
\[ \boxed{ F(x) = \cloze{\exp \left( \int^x \frac{
\beta - \alpha'}{\alpha} \dd x \right) } \tag{2.8}} \]
\end{flashcard}
and $(F\alpha y')' + F\gamma y = -\lambda F\rho
y$, so $p(x) = F(x) \alpha(x)$, $q(x) = -F(x)
\gamma(x)$ and $\omega(x) = F(x) \rho (x)$ (note
$F(x) > 0$).
    
\begin{example*}
    The Hermite equation for simple harmonic
    motion:
    \[ y'' - 2xy' + 2ny = 0 \]
    We want to put into Sturm-Liouville form
    (2.7). Comparing to (2.6) we have $\alpha =
    1$, $\beta = -2x$, $\gamma = 0$, $\lambda \rho
    = 2n$. By (2.8),
    \[ F = \exp \left( \int^x \frac{-2x - 0}{1}
    \dd x \right) = e^{-x^2} \]
    Hence
    \[ \mathcal{L} y \equiv -(e^{-x^2} y')' =
    2ne^{-x^2} y \tag{2.9} \]
\end{example*}

\begin{flashcard}
\begin{definition*}[Self-adjoint differential
    operator]
    $\mathcal{L}$ is self-adjoint on $a \le x \le
    b$ for all pairs of functions $y_1, y_2$
    satisfying
    \cloze{appropriate boundary conditions} if
    \[ \cloze{\langle y_1, \mathcal{L} y_2 \rangle =
    \langle \mathcal{L} y_1, y_2 \rangle} \]
    \fcscrap{or
    \[ \boxed{\int_a^b y_1^*(x) \mathcal{L}
    y_2(x) \dd x = \int_a^b (\mathcal{L} y_1(x))^*
    y_2(x) \dd x} \tag{2.10} \]}
\end{definition*}
\end{flashcard}

\noindent
Boundary conditions: substitute
Sturm-Liouville form (2.7) in (2.10) to find
\begin{align*}
    \langle y_1, \mathcal{L} y_2 \rangle - \langle
    \mathcal{L} y_1, y_2 \rangle
    &= \int_a^b [-y_1 (py_2')' + y_1 qy_2
    + y_2(py_1')' - y_2 qy_1] \dd x \\
    &= \int_a^b [-(py_1y_2')' + (py_1'y_2)'] \dd x
    \\
    &= [-py_1y_2' + py_1'y_2]_a^b
\end{align*}
and we want this to be 0 for given boundary
conditions at $x = a = b$. \myskip
Self-adjoint compatible boundary conditions
include:
\begin{itemize}
    \item Homogeneous $y(a) = y(b) = 0$ or $y'(a)
        = y'(b) = 0$ or mixed $y + ky' = 0$ (note
        regular Sturm-Liouville $\equiv$
        homogeneous boundary conditions)
    \item Periodic $y(a) = y(b)$
    \item Singular points of ODE $\rho(a) =
        \rho(b) = 0$
    \item Combinations of the above.
\end{itemize}

\subsection{Properties of self-adjoint operators}

\begin{enumerate}[(1)]
    \item Eigenvalues $\lambda_n$ are \emph{real}.
    \item Eigenfunctions $y_n$ are
        \emph{orthogonal}.
    \item Eigenfunctions $y_n$ form a
        \emph{complete set}.
\end{enumerate}
