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

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

2001-06-11

arxiv.org/abs/cond-mat/0106184v1

Physics

Condensed Matter

Materials Science

7 pages, 1 figure

Scientific paper

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 Schr\"{o}dinger equation at zero energy, when localized, lead to approximate upper bounds for critical screening lengths of the two-dimensional statically screened Coulomb potential.

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