Conditions for bound states of the pseudopotential with harmonic confinement in arbitrary dimensions

Abstract

We determine the conditions for bound states (E<0) for arbitrary Cartesian dimension d using a shape-independent regularized pseudopotential with scattering length a for two cold particles in a harmonic trap. It is known for d<=3 that the regularized pseudopotential supports one bound state for positive scattering length but does not support bound states for negative scattering length. We find that the usual (d<=3) positive scattering length bound states rule holds for certain higher odd dimensions d=4n+3 (n=0, 1, ...), but the existence of pseudopotential bound states at other odd dimensions requires a negative scattering length. Specifically, bound states are allowed in higher dimensions d=4n+1 (n=1, 2, ...) but they require a negative scattering length, which might suggest a universe in these dimensions might lead to different chemistry than d=3. We derive analytical approximations for bound state (E<0) and scattering (E>0) energies for a harmonic trap perturbed by the pseudopotential in arbitrary dimensions.