Lower limit in semiclassical form for the number of bound states in a central potential

Abstract

We identify a class of potentials for which the semiclassical estimate N (semi) =(1/π)∫ 0 ∞dr −V(r)θ[−V(r)] of the number N of (S-wave) bound states provides a (rigorous) lower limit: N⩾{{ N (semi)}}, where the double braces denote the integer part. Higher partial waves can be included via the standard replacement of the potential V( r) with the effective ℓ-wave potential V ℓ (eff)( r)= V( r)+ℓ(ℓ+1)/ r 2. An analogous upper limit is also provided for a different class of potentials, which is however quite severely restricted.