Math 6410 • ODE

Consider the differential operator \(Lu=-u''\) with boundary conditions \(u'(0)=u'(1)=0\).
\begin{enumerate}

\item
What Hilbert space and inner product is used for the analysis of this operator?


\item
Is this operator symmetric? Verify your answer.


\item
Is this operator invertible? Why or why not?


\item
What are the eigenvalues and eigenfunctions for this operator?


\item
Are these eigenfunctions complete on the domain of the operator? If yes, how do you
know (what theorem can you invoke?) and if no, produce a function in the domain of
L that is orthogonal to all of the eigenfunctions.


\item
Is \(f(x)=\cos^2(\pi x)\) in the domain of the operator? Find a representation of \(f(x)\) in
terms of the eigenfunctions of \(L\), if it exists?


\item
Is \(f(x)=\cos^2(\pi x)\) in the range of \(L\)? Why or why not?

\end{enumerate}