It is well-known that for usual Schrödinger 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 Schrödinger-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.

Joint work with Vu Hoang, Johanna Richter, and Semjon Wugalter