Quantitative bounds versus existence of weakly coupled bound states for Schr\"odinger type operators

It is well-known that for usual Schroedinger operators weakly coupled bound states exist in dimensions one and two, whereas in higher dimensions the famous Cwikel-Lieb-Rozenblum bound holds. We show for a large class of Schr\"odinger-type operators with general kinetic energies that these two phenomena are complementary. In particular, we explicitly get a natural semi-classical type bound on the number of bound states precisely in the situation when weakly coupled bound states exist not.