Supersymmetric WKB and WKB phase shifts for the Coulomb potential, the inverse-square potential, and their combinations

Abstract

In order to assess the accuracy of the supersymmetric WKB (SWKB) approximation for the calculation of phase shifts, the SWKB and WKB (with the Langer modification) phase shifts up to O(h 2 ) are compared numerically to the exact calculations for the Coulomb potential B/r, the inverse-square potential A/r 2 and their combinations A/r 2 +B/r, which include the Kratzer (A > 0, B < 0) and inverted Kratzer (A < 0, B > 0) potentials. For the inverse-square potential both the SWKB and the WKB methods yield the exact phase shifts. For the other two potentials considered, the results show that the SWKB phase shifts are in many cases superior; however, it is found that the = 0 WKB Coulomb phase shifts are more accurate for 0.8 < κ < 1.3 with κ = mB/h 2 k. In addition, the excellent agreement of the SWKB phase shifts up to O(h 4 ) with the exact ones shows that the SWKB series expansion in powers of h is rapidly convergent for the potentials considered.

Cold atoms at unitarity and inverse square interaction

“…Our purpose in this paper is to determine the statistical parameter g from the second virial coefficient. The second virial coefficient is obtained from the inverse square potential in terms of g using a semiclassical procedure, which is known to reproduce the exact quantum result [16]. Liu et al have also obtained it directly from the quantum spectrum given in figure 1.…”

An elementary exposition of the Efimov effect

“…−βV ℓ (r) dr. (A.3)The Laplace inversion of Z ℓ (β) with respect to β gives the density of statesg ℓ (l (r) Θ(E − V ℓ (r)). (A.4)For the inverse square interaction a WKB-type approximation yields exact results when the Langer correction is implemented,39 that is, ℓ(ℓ + 1) is replaced by (ℓ + 1/2) 2 . Hence, are interested in the ℓ = 0 partial wave, for which V 0 (r) = −h states is obtained by integrating Eq.…”

Cold atoms at unitarity and inverse square interaction

“…The only exception to this rule is the scale invariant inverse square potential. More over, when the Langer modification [16] of replacing l(l+1) by (l+1/2) 2 is implemented, the WKB approximation reproduces the quantum results exactly. As seen from the one dimensional example, the s-wave asymptotic wave function is exactly reproduced at resonance when the scattering length a → ∞.…”

“…Our purpose in this paper is to determine the statistical parameter g from the second virial coefficient. The second virial coefficient is obtained from the inverse square potential in terms of g using a semiclassical procedure, which is known to reproduce the exact quantum result [16]. Liu et.…”