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\centerline{Mathematics 174 -- Applied Mathematics 2}
\centerline{Discussion 1 -- Sturm-Liouville Eigenvalue Problems}
\centerline{January 20, 2000}
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\noindent
{\it Background}
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Last time, we introduced the standard Sturm-Liouville boundary
value problem:
$$\cases{{d\over dx}\left(p(x){d \phi(x)\over dx}\right) + q(x) \phi(x) + 
\lambda\sigma(x)\phi(x) = 0 & \cr
\beta_1 \phi(a) + \beta_2 \phi'(a) = 0 &\cr
\beta_3 \phi(b) + \beta_4 \phi'(b) = 0 &\cr}$$
We say the problem is {\it regular} if $p,q,\sigma$ are continuous
functions on $[a,b]$, and $p(x), \sigma(x) > 0$ for all $x \in [a,b]$.
For a regular Sturm-Liouville problem, we stated the following
results:
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\item{1.}  The set of eigenvalues (the set of all
constants $\lambda$ for which a non-zero solution $\phi(x)$ 
of the problem exists) forms a {\it sequence} of
real numbers.  This sequence 
has a smallest element $\lambda_1$, and can be listed in increasing
order:
$\lambda_1 < \lambda_2 < \lambda_3 < \cdots$.
When this is done, $\lambda_n/n \to \infty$ as $n \to \infty$ (so 
in particular, $\lambda_n$ grows faster than $n$ as $n\to \infty$).
\item{2.}  For each $n$, the eigenspace for $\lambda_n$ is 1-dimensional.
Call a basis eigenfunction $\phi_n(x)$.  $\phi_n(x)$
has exactly $n - 1$ distinct zeroes in the open interval $(a,b)$
(and possibly additional zeroes at the endpoints).
\item{3.}  The following orthogonality relation holds: If $m \ne n$
$$\int_a^b \phi_m(x) \phi_n(x) \sigma(x)\ dx = 0.$$
\item{4.}  The system of eigenfunctions $\{\phi_1(x),\phi_2(x),\ldots\}$
has the {\it completeness} property.  In particular,
if $f$ is a continuous
function which satisfies $\int_a^b f(x) \phi_n(x)\sigma(x)\ dx = 0$ for
all $n$, then $f(x)$ is identically zero on $[a,b]$.
\item{5.}  If $f(x)$ is piece-wise smooth on $[a,b]$, then we
can expand $f(x)$ using the eigenfunctions as
$f(x) \sim \sum_{n=1}^\infty a_n \phi_n(x)$
where the ``Fourier coefficients'' $a_n$ are given by:
$$a_n = {\int_a^b f(x) \phi_n(x) \sigma(x)\ dx \over
         \int_a^b (\phi_n(x))^2 \sigma(x)\ dx }.$$
Morover, as in the Pointwise
Convergence Theorem for Fourier Series,
this series converges to ${1\over 2}(f(x+) + f(x-))$
for all $x \in (a,b)$.
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\noindent
Today, we want to look at further examples, verify statements 1 - 3
(the rest are more difficult), and consider some ramifications of
these ideas.
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\noindent
{\it Discussion Questions}
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\item{A)}  Consider the Sturm-Liouville problem
$$\cases{{d^2\phi(x) \over dx^2} + \lambda \phi(x) = 0&\cr
         \phi'(0) = 0&\cr
         \phi(L) = 0&\cr}$$
Identify the functions $p(x),q(x),\sigma(x)$ here.
What are the eigenvalues and eigenfunctions for this problem?  Show
that all parts of statements 1,2,3 above are satisfied for the eigenvalues
and eigenfunctions.
\item{B)}  The second order ODE in the standard Sturm-Liouville
problem can be rewritten as
$${d\over dx}\left(p(x){d \phi(x)\over dx}\right) + q(x) \phi(x) =
- \lambda\sigma(x)\phi(x)\leqno(1)$$
Write $L$ for the {\it operator} on functions on the left side.
That is,
$$L(\phi) = {d\over dx}\left(p(x){d \phi\over dx}\right) + q(x) \phi.$$
Show that $L$ is a {\it linear operator} on functions.
\item{C)}  Consider the 2nd order linear differential operator
$$\overline{L}(\phi) = x^2 {d^2\phi\over dx^2} + x {d\phi\over dx}$$
Take the equation $\overline{L}(\phi) = -\lambda\phi$ and divide
both sides by $x$.  The result can be rearranged to Sturm-Liouville form:
$${d\over dx}\left(x {d\phi\over dx}\right) + {1\over x}\phi = -{\lambda\over x} \phi.$$
\itemitem{1)}  Suppose we add the boundary
conditions $\phi(1) = 0$, $\phi(2) = 0$.  Is this a {\it 
regular} Sturm-Liouville problem?  Why or why not?
\itemitem{2)}  Verify that $\lambda_n = \left({n\pi\over \ln2}\right)^2$
is an eigenvalue of this problem, with corresponding eigenfunction
$\phi_n(x) = \sin(\sqrt{\lambda_n}\ln(x))$.  
\itemitem{3)}  Verify that property 3 above (the orthogonality 
relation) is satisfied for the appropriate $\sigma(x)$.
\item{D)}  The Sturm-Liouville
operator $L$ in question B looks rather special; however 
we will see that in fact it is quite general.  An apparently more
general second-order linear differential operator $\overline{L}$
on functions would look like this:
$$\overline{L}(\phi) = {d^2\phi\over dx} + B(x) {d \phi\over dx} + 
C(x)\phi$$
The equation $\overline{L}(\phi) = -\lambda\phi$ is the eigenvalue/eigenvector
equation for the operator $\overline{L}$.
In general, show that if we multiply
both sides of the equation
$${d^2\phi\over dx^2} + B(x) {d \phi\over dx} + C(x)\phi = -\lambda \phi$$
by the function $e^{\int B(x)\ dx}$ (called an ``integrating factor''), then
the result can be rewritten in Sturm-Liouville form for 
suitable functions $p(x),q(x),\sigma(x)$ (what are they)?
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\noindent
{\it Assignment}
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Group write-ups, due in class on Thursday, January 27. 
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