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\def\Rl{\mathbb{R}}
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\centerline{{\sc Problem Set 6}}
\centerline{Math 207B, Winter 2012}
\centerline{Due: Fri., Feb. 24}


\bigskip\noindent
{\bf 1.} Suppose that $\lambda\in \Cx\setminus [0,\infty)$ is not a nonnegative
real number. Show that the Green's function for the BVP
\[
-u'' = \lambda u  + f(x),\qquad 0<x<\infty,
\qquad
u(0) = 0,\quad u\in L^2(0,\infty)
\]
is given by
\[
G(x,\xi;\lambda) = \frac{1}{i\sqrt{-\lambda}}\sin\left(i\sqrt{-\lambda}x_{<}\right) \exp\left(-\sqrt{-\lambda} x_{>}\right)
\]
where $\sqrt{-\lambda}$ is the branch of the square root with positive real part and
\[
x_{<} = \min (x,\xi),\qquad x_{>} = \max(x,\xi).
\]
What singularities does $G$ have as a function of $\lambda$? Write down the Green function representation for the solution
of the BVP.


\bigskip\noindent
{\bf 2.} Consider the Sturm-Liouville problem
\[
-u'' = \lambda u \qquad 0<x<\infty,
\qquad
(\cos \alpha) u(0) - (\sin \alpha) u'(0) = 0
\]
where $0 \le \alpha\le \pi$ is a real constant.
Show that this has an eigenfunction
$u\in L^2(0,\infty)$ if and only if $-\pi/2 < \alpha < \pi$,
and in that case $\lambda = -\cot^2\alpha$.
(This problem also has a continuous spectrum with $0\le \lambda < \infty$, similar to the one in Problem 1.)


\bigskip\noindent
{\bf 3.} Consider the Sturm-Liouville eigenvalue problem
\[
-(x^2 u')' = \lambda u \qquad 1<x<e,
\qquad
u(1) = 0,\qquad u(e) =0.
\]
Is it regular or singular? Show that the eigenvalues and eigenfunctions are given by
\[
\lambda_n = n^2\pi^2 + \frac{1}{4},\qquad u_n(x) = x^{-1/2} \sin \left(n\pi \log x\right).
\]
Write out the corresponding eigenfunction expansion of a function $f\in L^2(1,e)$.

\smallskip\noindent
\textsc{Hint.} Note that the ODE is an Euler equation and look for solutions of the form $u(x) = x^r$.

\bigskip\noindent
{\bf 4.} Consider the singular Sturm-Liouville eigenvalue problem for Legendre's equation
\begin{align*}
&-[(1-x^2) u']' = \lambda u \qquad -1 < x < 1,
\\
&(1-x^2)u'(x) \to 0 \quad \mbox{as $x\to \pm 1$}.
\end{align*}
(a) Solve the ODE for $\lambda=0$ and show that both endpoints $x=\pm 1$ are in the limit circle case.

\smallskip\noindent
(b) For $n=0,1,2,\dots$, define the Legendre polynomials $P_n(x)$ by
\[
P_n(x) = \frac{1}{2^n n!}\frac{d^n}{dx^n} \left[(x^2-1)^n\right].
\]
(Note that $P_n$ \emph{is} a polynomial of degree $n$.)
Show that the Legendre polynomials are eigenfunctions of the Legendre equation with eigenvalues
\[
\lambda_n = n(n+1).
\]
\textsc{Hint.} Let $v(x) = (x^2-1)^n$ and differentiate the equation
$(x^2-1)v' = 2n x v$
$n+1$ times.

\smallskip\noindent
(c) With $v$ as in (b), show that
\[
\int_{-1}^1 \left[\frac{d^nv}{dx^n}\right]^2 = (2n)! \int_{-1}^1 (1-x)^n (1+x)^n\, dx
= \frac{(n!)^2}{(2n)!} \int_{-1}^1 (1+x)^{2n}\, dx
\]
and deduce that
\[
\int_{-1}^1 P_n(x)^2 \, dx = \frac{2}{2n+1}.
\]
(d) 
Write out the orthogonality relations for the Legendre polynomials and the
eigenfunction expansion of a function $f\in L^2(-1,1)$ with respect to the Legendre polynomials.
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