We calculate the number of bound states appealing below the spectrum of a semi-bounded operator in the case of a weak, non-signdefined perturbation. The abstract result generalizes the Birman-Schwinger principle to this case. We discuss a number of examples, in particular higher order differential operators, critical Schrodinger operators, systems of second order differential operators, Schrodinger type operators with magnetic fields and the two-dimensional Pauli operator with localized magnetic field.
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