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Analytical calculations of scattering lengths for a class of long-range potentials of interest for atomic physics

Abstract

We derive two equivalent analytical expressions for an $l$th partial-wave scattering length $a_{l}$ for central potentials with long-range tails of the form % \begin{math} \displaystyle V(r)=-\frac{\hbar^{2}}{2m}\frac{Br^{n-4}}{(r^{n-2}+R^{n-2})^{2}} -\frac{\hbar^{2}}{2m}\frac{C}{r^{2}(r^{n-2}+R^{n-2})}, \end{math} % ($r\geqslant r_{s}$, $R>0$). % For $C=0$, this family of potentials reduces to the Lenz potentials discussed in a similar context in our earlier works [Acta Phys. Pol.\ A 79 (1991) 613 and J.\ Phys.\ A 28 (1995) 7333]. The formulas for $a_{l}$ which we provide in this paper depend on the parameters $B$, $C$ and $R$ characterizing the tail of the potential, on the core radius $r_{s}$, as well as on the short-range scattering length $a_{ls}$, the latter being due to the core part of the potential. The procedure, which may be viewed as an analytical extrapolation from $a_{ls}$ to $a_{l}$, is relied on the fact that the general solution to the zero-energy radial Schr{\"o}dinger equation with the potential given above may be expressed analytically in terms of the \emph{generalized\/} associated Legendre functions.

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Details

Category:
Articles
Type:
artykuły w czasopismach
Published in:
JOURNAL OF MATHEMATICAL PHYSICS no. 61,
ISSN: 0022-2488
Language:
English
Publication year:
2020
Bibliographic description:
Szmytkowski R.: Analytical calculations of scattering lengths for a class of long-range potentials of interest for atomic physics// JOURNAL OF MATHEMATICAL PHYSICS -Vol. 61,iss. 1 (2020), s.012103-
DOI:
Digital Object Identifier (open in new tab) 10.1063/1.5140726
Verified by:
Gdańsk University of Technology

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