Counting the Number of Bound States of Two-dimensional Screened Coulomb Potentials: A Semiclassical Approach

Abstract

Portnoi and Galbraith recently proposed a beautiful and intriguing relationship defining the critical screening lengths associated with the apparition of new bound states for the two-dimensional statically screened Coulomb potential. Not only does semiclassical quantum theory show that this relationship is unfortunately not strictly exact, it has also proved helpful in the search for a potential which exactly verifies Portnoi and Galbraith's formula, namely $\frac{-e^2}{r(1+r/r_s{)}^2}$. The analytical eigenfunctions of the corresponding Schrödinger equation at zero energy, when localized, lead to approximate upper bounds for critical screening lengths of the two-dimensional statically screened Coulomb potential.

arXiv comment7 pages, 1 figure