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\centerline{{\sc Problem set 1}}
\centerline{Math 207B, Winter 2012}
\centerline{Due: Fri., Jan. 20}


\bigskip\noindent
{\bf 1.} If $\lambda \le 0$, show that there are no non-zero solutions for $u(x)$ of the Sturm-Liouville problem
\begin{align*}
&- u'' = \lambda u\qquad 0<x<1,
\\
&u(0)=0,\qquad u(1) = 0.
\end{align*}

\bigskip\noindent
{\bf 2.}  Find all eigenvalues $\lambda_n$ and eigenfunctions $u_n(x)$  of the Sturm-Liouville problem
\begin{align*}
&- u'' = \lambda u\qquad 0<x<1,
\\
&u'(0)=0,\qquad u'(1) = 0
\end{align*}
with Neumann boundary conditions. Show that
\[
\int_0^1 u_n(x) u_m(x)\, dx = 0
\]
if $n\ne m$.


\bigskip\noindent
{\bf 3.} Consider a wave equation
\[
u_{tt} - \left(c_0^2 u_x\right)_x + q_0 u = 0
\]
for $u(x,t)$ where the wave speed $c_0(x)$ and the coefficient $q_0(x)$ depend on $x$. Look for (possibly complex-valued)
separable solutions of the form
\[
u(x,t) = X(x) T(t).
\] 
Show that, up to linear superpositions, we can take
\[
T(t) = e^{-i\omega t}
\]
for some constant $\omega \in \Cx$. Find the corresponding ODE satisfied by $X$
in that case. (Note that if a function of $x$ is equal to a function
of $t$, then the functions must be constant.)



\bigskip\noindent
{\bf 4.}
Define the Hermite polynomials $H_n$ by
\[
H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n}\left(e^{-x^2}\right)
\]
and let
\[
\phi_n(x) = e^{-x^2/2}H_n(x)
\]
where $n=0,1,2,3,\dots$.
Show that $\phi_n$ is a solution of the Sturm-Liouville problem
\begin{eqnarray*}
&&-\phi_n'' + x^2 \phi_n = \lambda_n \phi_n \qquad -\infty < x < \infty,
\\
&&\phi_n(x) \to 0\quad\mbox{as $|x|\to \infty$}
\end{eqnarray*}
with eigenvalue
\[
\lambda_n = 2n + 1.
\]
{\sc Hint}: Let $L$ be the linear operator
\[
L = -\frac{d^2}{dx^2} + x^2,
\]
meaning that $L$ acts on functions $\phi$ by
\[
L \phi = - \phi'' + x^2 \phi.
\]
Define operators $A$, $A^\ast$ by
\[
A = \frac{d}{dx} + x,\qquad A^\ast = -\frac{d}{dx} + x .
\]
Show that
\[
A\phi_n = 2n \phi_{n-1}, \quad A^\ast \phi_n = \phi_{n+1}
\]
and
\[
L = A A^\ast - 1.
\]
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