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 $N^{\text{(semi)}}=\frac{1}{\pi }{\int }_0^{\infty }dr\sqrt{-V(r)\theta [-V(r)]}$ of the number $N$ of (S-wave) bound states provides a (rigorous) lower limit: $N\ge N^{\text{(semi)}}$, 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 $\ell$-wave potential $V_{\ell }^{\text{(eff)}}(r)=V(r)+\frac{\ell (\ell +1)}{r^2}$. An analogous upper limit is also provided for a different class of potentials, which is however quite severely restricted.

9 pages

References in corpus (3)

• Upper and lower limits for the number of S-wave bound states in an attractive potential
• Upper and lower limits on the number of bound states in a central potential
• A class of ($\ell$-dependent) potentials with the same number of ($\ell$-wave) bound states

Cited by in corpus (3)

• Transition states and the critical parameters of central potentials
• Upper limit on the critical strength of central potentials in relativistic quantum mechanics
• Upper and lower bounds on the mean square radius and criteria for occurrence of quantum halo states