First at all I have to say that I've done this questions a couple of hours ago in the Math section of this page, but I think that there could possible be more answers here then there because I'm actually reading these topics on a book of Methods of Modern Mathematical Physics.
I've recently started studying some of perturbation theory and I've noted that there are several operators (unitarily equivalent to the position operator in deed) that doesn't have any eigenvalue in their spectrum, but by adding little perturbations you can add even countables of them. However, the analysis is interesting by itself, but I would like to know more about applications, 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 thing happens when you take the Heat equation on bounded domains.
So the question is: Do you know any other examples of problems on operators that are solvables by knowing eigenvalues/vectors? I hope the question is not so open.