Math 6410 • ODE

Define a regular Sturm-Liouville operator by the equation
\[
\calL u=-(2x)^{-1}\bigl((2x)^{-1}u'\bigr)'
\]
on \(1\le x\le\sqrt{2}\) using boundary conditions \(u(1)=u(\sqrt{2})=0\).

\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 \(S\) using the solution basis
\(u_1(x)=\sin(\sqrt{\lambda}x^2)\), \(u_2(x)=\cos(\sqrt{\lambda}x^2)\) of the differential
equation \(\calL u-\lambda u=0\). Do not derive the basis.

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