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

In a recent paper new upper and lower limits were given, in the context of the Schrödinger 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 N of bound states for the th 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 limitN {{[σ/(2 +1)+1]/2}}with σ = (2/π)max0 r<∞[r|V (r)|1/2] (valid in the Schrödinger case, for a class of potentials that includes the monotonically nondecreasing ones), entailing the following lower limit for the total number N of bound states possessed by a monotonically nondecreasing central potential vanishing at infinity: N {{(σ +1)/2}}{{(σ +3)/2}}/2 (here the double braces denote the integer part). PACS numbers: 03.65.−w, 03.65.Ge