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

Fabian Brau·Francesco Calogero
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Abstract

We identify a class of potentials for which the semiclassical estimate $N^{\text{(semi)}}=\frac{1}π\int_0^\infty dr\sqrt{-V(r)θ[-V(r)]}$ 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.

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Fabian Brau, Francesco Calogero. 2004-02-10. Lower limit in semiclassical form for the number of bound states in a central potential. https://doi.org/10.1016/j.physleta.2003.12.034

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