Exact Results in Quantum Theory

The present talk is divided in two parts: The first deals with Sturm--Liouville operators that have trace class resolvents. Our main finding is the introduction of the \emph{regularization index} $\ell \in\mathbb N \cup \{ \infty \}$ which extends the limit circle/limit point dichotomy in the sense that $\ell = 0$ at some endpoint if and only if the expression is in the limit circlecase. In the limit point case $\ell > 0$, we present a natural interpretation of $\ell$ interms of iterated Darboux transforms. The second part deals with the general Schatten $p$-class resolvent case. We show that the growing/decaying solutions near the endpoints display universal singular behavior. Our main application of this result is related to spectral $\zeta$-functions, in particular contour integral representations, which are used to compute their analytic continuation. This is joint work with Guglielmo Fucci from East Carolina University and Jonathan Stanfill from Ohio State University.