1. Analytical calculations of scattering lengths for a class of long-range potentials of interest for atomic physics.
- Author
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Szmytkowski, Radosław
- Subjects
ATOMIC physics ,LEGENDRE'S functions ,SCHRODINGER equation ,SCATTERING (Mathematics) ,MATHEMATICS ,EXTRAPOLATION ,SCHRODINGER operator - Abstract
We derive two equivalent analytical expressions for an lth partial-wave scattering length a
l for central potentials with long-range tails of the form V (r) = − ℏ 2 2 m B r n − 4 (r n − 2 + R n − 2) 2 − ℏ 2 2 m C r 2 (r n − 2 + R n − 2) , (r ⩾ rs , R > 0). For C = 0, this family of potentials reduces to the Lenz potentials discussed in a similar context in our earlier works [R. Szmytkowski, Acta Phys. Pol. A 79, 613 (1991); J. Phys. A: Math. Gen. 28, 7333 (1995)]. The formulas for al that we provide in this paper depend on the parameters B, C, and R characterizing the tail of the potential, on the core radius rs , as well as on the short-range scattering length als , the latter being due to the core part of the potential. The procedure, which may be viewed as an analytical extrapolation from als to al , is relied on the fact that the general solution to the zero-energy radial Schrödinger equation with the potential given above may be expressed analytically in terms of the generalized associated Legendre functions. [ABSTRACT FROM AUTHOR]- Published
- 2020
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