Generalized Pseudopotentials for Higher Partial Wave Scattering

Abstract

We derive a generalized zero-range pseudopotential applicable to all partial wave solutions to the Schrödinger equation based on a delta-shell potential in the limit that the shell radius approaches zero. This properly models all higher order multipole moments not accounted for with a monopolar delta function at the origin, as used in the familiar Fermi pseudopotential for s-wave scattering. By making the strength of the potential energy dependent, we derive self-consistent solutions for the entire energy spectrum of the realistic potential. We apply this to study two particles in an isotropic harmonic trap, interacting through a central potential, and derive analytic expressions for the energy eigenstates and eigenvalues.

Revised Huang-Yang multipolar pseudopotential

“…As discussed in the introduction there is some evidence from the literature [4,5] that there is a difficulty with the original HY formula. Below I provide an alternative self-consistency check of the HY pseudopotential based on the integral equation for the phase shifts.…”

“…Second, an alternative derivation of the pseudopotential has been presented by Stock et al [5]. Instead of deltafunction lumped at the origin, these authors have proposed to use a shell pseudo-potential, i.e., delta-function placed at the surface of spherical shell.…”

“…Because of these missing factors, the authors of Ref. [5] claimed that there is a "fundamental problem" with the Huang-Yang derivation.…”

“…Notice that the s-wave contribution (l = 0) to V ps is not affected by the correction. Since the s waves dominate lowenergy collision physics, certain inconsistencies for higher partial waves has not been noticed until very recently [4,5], when higher order multipoles became a subject of interest. For example, for identical fermions the s-wave contribution vanishes because of the symmetry arguments, and one has to consider p-wave scattering in particular.…”

Optical Feshbach resonances of alkaline-earth-metal atoms in a one- or two-dimensional optical lattice

“…All these works relate the effective 1D or 2D scattering properties to the free-space properties. Also closely related are the constructions of pseudo-potentials which can treat confined systems [21,22,23] and pseudo-potentials in low-dimensional systems [22,24].…”