Rapid, accurate calculation of the s-wave scattering lengthстатья
Статья опубликована в высокорейтинговом журнале
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Аннотация:Transformation of the conventional radial Schroumldinger equation
defined on the interval r is an element of {[}0, infinity) into an
equivalent form defined on the finite domain y(r) is an element of {[}a,
b] allows the s-wave scattering length a(s) to be exactly expressed in
terms of a logarithmic derivative of the transformed wave function
phi(y) at the outer boundary point y = b, which corresponds to r =
infinity. In particular, for an arbitrary interaction potential that
dies off as fast as 1/r(n) for n >= 4, the modified wave function phi(y)
obtained by using the two-parameter mapping function r(y;(r) over bar,
beta)=(r) over bar {[}1+ 1/beta tan(pi y/2)] has no singularities, and
a(s)=(r) over bar inverted right perpendicular1 + 2/pi beta 1/phi(1) d
phi(1)/dyinverted left perpendicular. For a well bound potential with
equilibrium distance r(e), the optimal mapping parameters are (r) over
bar approximate to r(e) and beta approximate to n/2 - 1. An outward
integration procedure based on Johnson's log-derivative algorithm {[}J.
Comp. Phys. 13, 445 (1973)] combined with a Richardson extrapolation
procedure is shown to readily yield high precision a(s)-values both for
model Lennard-Jones (2n, n) potentials and for realistic published
potentials for the Xe-e(-), Cs(2)(a(3)Sigma(+)(u)), and (3,
4)He(2)(X(1)Sigma(+)(g)) systems. Use of this same transformed
Schroumldinger equation was previously shown {[}V. V. Meshkov , Phys.
Rev. A 78, 052510 (2008)] to ensure the efficient calculation of all
bound levels supported by a potential, including those lying extremely
close to dissociation.