Most of what I do is related, in one way or another, to the equation -y''(x)+V(x)y(x)=Ey(x). This is the (time-independent) Schrödinger equation in one space dimension. I've always found it fascinating to see how this innocuous looking equation can give rise to such a rich theory that draws from almost all core areas of classical mathematical analysis.
Click here for my preprints and recent (and not so recent) papers on the subject. I have also compiled seven Millennium Prize Problems on Schrödinger operators (mainly in one space dimension). I'm now thinking about (partially) replacing these by new questions. While the original problems still seem to be open, with the exception of problem #6, they don't reflect my current view of the subject very well any more.