If \(A\) is a matrix, then the eigenvalues of \(A\) tell us something about the stability of the ODE \(\dot u = A u\text{.}\) Now, if \(A\) is an unbounded operator, (e.g. \(A=d\Delta\)), what information can we get from the spectrum of \(A\text{?}\) What actually is the spectrum of \(A\) in that case? We will see in the chapter on spectral theory (Chapter Chapter 8) that the spectrum of \(A\text{,}\) denoted as \(\sigma(A)\) might contain much more than just eigenvalues.
