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\centerline{{\sc Problem set 2}}
\centerline{Math 207B, Winter 2012}
\centerline{Due: Fri., Jan. 27}


\bigskip\noindent
{\bf 1.} Suppose that $L$ is the second-order differential operator
\[
L = p_2(x) \frac{d^2}{dx^2} + p_1(x) \frac{d}{dx} + p_0(x)
\]
Find the adjoint operator
\[
L^\ast = q_2(x) \frac{d^2}{dx^2} + q_1(x) \frac{d}{dx} + q_0(x)
\]
such that
\[
\int_a^b \left(u L^\ast v - v L u\right)\, dx = 0
\]
for all functions $u, v : [a,b]\to \Rl$ that satisfy
\[
u(a) = u'(a) = 0,\ u(b) = u'(b) = 0,\quad
v(a) = v'(a) = 0,\ v(b) = v'(b) = 0.
\]
Show that the Sturm-Liouville operator
\[
L = - \frac{d}{dx} p(x) \frac{d}{dx} + q(x)
\] 
is the most general formally self-adjoint operator (meaning that $L=L^\ast$).

\bigskip\noindent
{\bf 2.} Consider the Sturm-Liouville operator
\[
L = - \frac{d}{dx} p(x) \frac{d}{dx} + q(x)
\]
acting on functions $u(x)$ defined on the interval $a\le x \le b$. Find adjoint boundary conditions in the following cases.
(a) Neumann $u'(a) = u'(b) =0$; (b) Periodic $u(a) = u(b)$, $u'(a) = u'(b)$; (c) Initial $u(a) = u'(a) = 0$.
Which boundary conditions are self-adjoint?

\newpage\noindent
{\bf 3.} If $u_1$, $u_2$ are two solutions of the Sturm-Liouville equation
\[
-(pu')' + q u = \lambda u
\]
show that
\[
p\left(u_1 u_2' - u_2 u_1'\right) = \mathrm{constant}.
\]
Deduce that if $p(x) > 0$ on $[a,b]$ and $u_1$, $u_2$ satisfy the separated BCs
\[
(\cos \alpha) u(a) + (\sin \alpha) u'(a) = 0,\qquad
(\cos \beta) u(b) + (\sin \beta) u'(b) = 0
\]
for some constants $\alpha$, $\beta$
then $u_1$, $u_2$ are linearly dependent.
(It follows that all eigenvalues are simple.) 


\bigskip\noindent
{\bf 4.} Find the eigenvalues and eigenfunctions of the Sturm-Liouville problem
\begin{align*}
& -u'' = \lambda u\qquad 0< x < 2\pi,
\\
&u(0) = u(2\pi),\qquad u'(0) = u'(2\pi)
\end{align*}
with periodic boundary conditions. Verify explicitly that eigenfunctions with different eigenvalues are
orthogonal. Are the eigenvalues simple? Is this answer consistent with the result of Problem 3?
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