I've recently started studying some of perturbation theory and I've noted that there are several operators that doesn't have any eigenvalue in their spectrum, but by adding little perturbations you can add even countable of them. The analysis is interesting by itself, I would like to know more about applications though. I know that somewhere there have to be plenty of them.
I've actually seen some examples in my PDE course, for example the Laplace's problem on certain domains is transformed into a fourier series problem by just knowing the eigenvalues/vectors, and the same happen when you take the heat equation on bounded domains.
So the question is: Do you know any other examples of problems with operators that are easier to solve by knowing their eigenvalues/vectors? I hope the question is not too open.