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

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, horizontal ellipsis ), 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, horizontal ellipsis ) but they require a negative scattering length, which suggests 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.