  We consider perturbations of Hamiltonians whose Fourier symbol attains its
minimum along a hypersurface. Such operators arise in several domains, like
spintronics, theory of supercondictivity, or theory of superfluidity.
Variational estimates for the number of eigenvalues below the essential
spectrum in terms of the perturbation potential are provided. In particular, we
provide an elementary proof that negative potentials lead to an infinite
discrete spectrum.
