Math 6410 • ODE

Define a regular Sturm-Liouville operator by the equation
\[
\calL u=-x^2(x^2u')'
\]
on \(1/2\le x\le1\) using boundary conditions \(u(1/2)=u(1)=0\). Complete the
following:

\begin{enumerate}
\item[(a)] Explain how \(\calL\) and the boundary conditions can be used to construct a
selfadjoint operator \(S\) in a suitable Hilbert space \(\calH\). Do not give proofs.

\item[(b)] Calculate the eigenvalues and eigenfunctions of the selfadjoint operator
\(S\), using the solution basis \(u_1(x)=\sin(\lambda/x)\), \(u_2(x)=\cos(\lambda/x)\) of
the differential equation
\[
x^2(x^2u')'+\lambda u=0.
\]
Do not derive the basis.

\item[(c)] Verify by direct integration that the eigenfunctions are orthogonal in the
Hilbert space \(\calH\).
\end{enumerate}