Upper and lower limits on the number of bound states in a central potential

Abstract

In a recent paper new upper and lower limits were given, in the context of the Schródinger or Klein-Gordon equations, for the number N0 of S-wave bound states possessed by a monotonically nondecreasing central potential vanishing at infinity. In this paper these results are extended to the number Nℓ of bound states for the lth partial wave, and results are also obtained for potentials that are not monotonic and even somewhere positive. New results are also obtained for the case treated previously, including the remarkably neat lower limit Nℓ ≥ {{[σ/(2l+1)+1]/2}}with σ = (2/π)max 0≤r