Generalization of the Birman-Schwinger method for the number of bound states

We generalize the Birman-Schwinger method, and derive a general upper bound on the number of bound states in the S wave for a spherically symmetric potential. This general bound includes, of course, the Bargmann bound, but also leads, for increasing (negative) potentials, to a Calogero-Cohn-type bound. Finally, we show that for a large class among these potentials, one can obtain further improvements. (C) 1999 American Institute of Physics. [S0022-2488(99)00804- X].