  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.
