Abstract More than 65 years ago, Jost and Kohn (1952 Phys. Rev. 87 , 977) derived an explicit expression for a class of short-range model potentials from a given effective range expansion with the s -wave scattering length a s being negative. For a s > 0, they calculated another class of short-range model potentials (Jost and Kohn 1953 Dan. Mat. Fys. Medd 27 , no. 9) using a method based on an adaptation from Gel’fand−Levitan theory (Gel’fand and Levitan 1951 Dokl. Akad. Nauk. USSR 77 , 557-60) of inverse scattering. We here revisit the methods of Jost and Kohn in order to explore the possibility of modeling resonant finite-range interactions at low energy. We show that the Jost−Kohn potentials can account for zero-energy resonances. The s -wave phase shift for positive scattering length is expressed in an analytical form as a function of the binding energy of a bound state. We show that, for small binding energy, both the scattering length and the effective range are strongly influenced by the binding energy; below a critical binding energy the effective range becomes negative provided the scattering length is large. As a consistency check we carry out some simple calculations to show that Jost−Kohn potentials can reproduce the standard results of contact interaction in the limit of the effective range going to zero.
We show that it is possible to model two-body resonant interactions at low energy with a class of finite-range potentials based on the methods of Jost and Kohn. These potentials are expressed in terms of the effective range [Formula: see text] and the [Formula: see text]-wave scattering length [Formula: see text]. We derive continuum solutions of these potentials. By writing [Formula: see text], where the sign [Formula: see text] refers to positive(negative) scattering length, [Formula: see text] is of the form of Pöschl–Teller potential and [Formula: see text] is expressed as a power series of the small parameter [Formula: see text] when [Formula: see text] is large, we derive Green’s function of [Formula: see text]. Using the Green’s function, solutions of [Formula: see text] for [Formula: see text] can be obtained numerically by treating [Formula: see text] as a perturbation. We describe the threshold behavior of scattering phase shift for [Formula: see text]. This study may be important for developing a better understanding of physics of strongly interacting ultracold atomic gases with tunable interactions.
The interaction between atoms behaves as $\ensuremath{-}\ensuremath{\alpha}{/r}^{n}$ at large distances and, owing to the large reduced mass \ensuremath{\mu} of the collision pair, allows a semiclassical treatment within the potential well. As a result, the low-energy scattering is governed by two large parameters: the asymptotic parameter $\ensuremath{\gamma}=\sqrt{2\ensuremath{\mu}\ensuremath{\alpha}}/\ensuremath{\Elzxh}\ensuremath{\gg}{a}_{0}^{(n\ensuremath{-}2)/2}$ (
a0 is the Bohr radius) and the semiclassical zero-energy phase $\ensuremath{\Phi}\ensuremath{\gg}1.$ In our previous work [Phys. Rev. A 48, 546 (1993)] we obtained an analytical expression for the scattering length
a, which showed that it has 75% preference for positive values for
n=6, characteristic of collisions between ground-state neutral atoms. In this paper we calculate the effective range and show that it is a function of
a, ${r}_{e}{=F}_{n}\ensuremath{-}{G}_{n}{/a+H}_{n}{/a}^{2},$ where
Fn, Gn, and
Hn depend only on \ensuremath{\gamma}. Thus, we know the
s-phase shift at low momenta $k\ensuremath{\ll}{\ensuremath{\gamma}}^{\ensuremath{-}2/(n\ensuremath{-}2)}$ from the expansion $k\mathrm{cot}{\ensuremath{\delta}}_{0}\ensuremath{\simeq}\ensuremath{-}1/a+\frac{1}{2}{r}_{e}{k}^{2}.$ At $k\ensuremath{\gg}{\ensuremath{\gamma}}^{\ensuremath{-}2/(n\ensuremath{-}2)}$ the phase shift is obtained semiclassically as ${\ensuremath{\delta}}_{0}=\ensuremath{\Phi}+\ensuremath{\pi}/4\ensuremath{-}{I}_{n}{\ensuremath{\gamma}}^{2/n}{k}^{(n\ensuremath{-}2)/n},$ where ${I}_{n}=[n/(n\ensuremath{-}2)]\ensuremath{\Gamma}((n\ensuremath{-}1)/n)\ensuremath{\Gamma}((n+2)/2n)/\sqrt{\ensuremath{\pi}}.$ Therefore, \ensuremath{\gamma} and \ensuremath{\Phi} determine the
s-wave atomic scattering in a wide range of momenta, as well as the positions of upper bound states of the diatomic molecule.
Abstract We calculate Bardeen–Cooper–Schrieffer (BCS) state of a unitary Fermi gas of atoms interacting with the finite-ranged Jost-Kohn potential which has been recently shown to account for the resonant interactions (2019 J. Phys. B: At. Mol. Opt. Phys. 52 165004). Using exact scattering solution of the potential, we derive two-body T-matrix element which is employed to construct the BCS Hamiltonian in momentum space. We present results on the energy- and range-dependence of the pairing gap and superfluid density and the range-dependence of the chemical potential for a wide variation of the scattering length including the unitary regime. In the zero range limit our calculated gap at the Fermi energy is found to be nearly equal to that calculated in mean-field theory with contact potential. The mean gap averaged over the full width at half maximum of the gap function in the zero range and unitary limits is found to be 0.42E F which is quite close to the recent result of the quantum Monte-Carlo simulation (2018 Phys. Rev. A 97 013601). The chemical potential in the zero range limit also agrees well with that for the contact potential.
Studies of cold atom collisions and few-body interactions often require the energy dependence of the scattering phase shift, which is usually expressed in terms of an effective-range expansion.We use accurate coupled-channel calculations on 6 Li, 39 K, and 133 Cs to explore the behavior of the effective range in the vicinity of both broad and narrow Feshbach resonances.We show that commonly used expressions for the effective range break down dramatically for narrow resonances and near the zero crossings of broad resonances.We present an alternative parametrization of the effective range that is accurate through both the pole and the zero crossing for both broad and narrow resonances.However, the effective-range expansion can still fail at quite low collision energies, particularly around narrow resonances.We demonstrate that an analytical form of an energy and magnetic-field-dependent phase shift, based on multichannel quantum defect theory, gives accurate results for the energy-dependent scattering length.
Positronium-hydrogen (Ps-H) scattering is of interest, as it is a fundamental four-body Coulomb problem. We have investigated low-energy Ps-H scattering below the Ps(n=2) excitation threshold using the Kohn variational method and variants of the method with a trial wavefunction that includes highly correlated Hylleraas-type short-range terms. We give an elegant formalism that combines all Kohn-type variational methods into a single form. Along with this, we have also developed a general formalism for Kohn-type matrix elements that allows us to evaluate arbitrary partial waves with a single codebase. Computational strategies we have developed and use in this work will also be discussed.With these methods, we have computed phase shifts for the first six partial waves for both the singlet and triplet states. The 1S and 1P phase shifts are highly accurate results and could potentially be viewed as benchmark results. Resonance positions and widths for the 1S-, 1P-, 1D-, and 1F-waves have been calculated.We present elastic integrated, elastic differential, and momentum transfer cross sections using all six partial waves and note interesting features of each. We use multiple effective range theories, including several that explicitly take into account the long-range van der Waals interaction, to investigate scattering lengths for the 1,3S and 1,3P partial waves and effective ranges for the 1,3S-wave.
We calculate Bardeen-Cooper-Schrieffer (BCS) state of a unitary Fermi gas of atoms interacting with the finite-ranged Jost-Kohn potential which has been recently shown to account for the resonant interactions [2019 {\rm J. Phys. B: At. Mol. Opt. Phys.} {\bf 52}, 165004]. Using exact scattering solution of the potential, we derive two-body
T-matrix element which is employed to construct the BCS Hamiltonian in momentum space. We present results on the energy- and range-dependence of the pairing gap and superfluid density and the range-dependence of the chemical potential for a wide variation of the scattering length including the unitarity regime. In the zero range limit our calculated gap at the Fermi energy is found to be nearly equal to that calculated with contact potential. The mean gap averaged over the full width at half maximum of the gap function in the zero range and unitarity limits is found to be
0.42EF which is quite close to the recent result of the quantum Monte Carlo simulation [2018 {\rm Phys. Rev.A} {\bf 97}, 013601]. The chemical potential in the zero range limit also agrees well with that for the contact potential.