Math 6410 • ODE

Consider the regular Sturm-Liouville operator
\[
\calL u=-e^{-2x}(e^{2x}u')'-u
\]
on \(0\le x\le\pi\) with boundary conditions \(u(0)=0\), \(u(\pi)=0\).

\begin{enumerate}
\item[(a)] Show how \(\calL u\) and the boundary conditions can be used to construct a
selfadjoint operator \(S\) in a suitable Hilbert space \(\calH\).

\item[(b)] Calculate the eigenvalues and the corresponding eigenfunctions of \(S\).
\end{enumerate}