Remarks on virtual bound states for semi-bounded operators

We calculate the number of bound states appearing below the spectrum of a s emi-bounded operator in the case of a weak, indefinite perturbation. The ab stract result generalizes the Birman-Schwinger principle to this case. We d iscuss a number of examples, in particular higher order differential operat ors, critical Schrodinger operators, systems of second order differential o perators, Schrodinger type operators with magnetic fields and the two-dimen sional Pauli operator with a localized magnetic field.