We present a multiscale quantum-defect theory for two identical atoms in a symmetric harmonic trap that combines the quantum-defect theory for the van der Waals interaction [B. Gao, Phys. Rev. A 64, 010701(R) (2001)] at short distances with a quantum-defect theory for the harmonic trapping potential at large distances. The theory provides a systematic understanding of two atoms in a trap, from deeply bound molecular states and states of different partial waves, to highly excited trap states. It shows, e.g., that a strong
p-wave pairing can lead to a lower energy state around the threshold than a
s-wave pairing.
We derive analytical solutions for the system of two ultracold spin-polarized fermions interacting in
p wave and confined in an axially symmetric harmonic trap. To this end we utilize
p-wave pseudopotential with an energy-dependent scattering volume. This allows to describe the scattering in tight trapping potentials in the presence of scattering resonances. We verify predictions of the pseudopotential treatment for some model interaction potential obtaining an excellent agreement with exact energy levels. Then we turn to the experimentally relevant case of neutral atom interactions in the vicinity of a
p-wave Feshbach resonance. In the framework of the multichannel quantum-defect theory we derive relatively simple formula for an energy-dependent scattering volume, and later we apply it to investigate the energy spectrum of trapped atoms close to the
p-wave Feshbach resonance.
Using the method of quantum-defect theory, we calculate the ultra-long-range molecular vibrational states near the dissociation threshold of a diatomic molecular potential which asymptotically varies as $\ensuremath{-}{1/R}^{3}.$ The properties of these states are of considerable interest as they can be formed by photoassociation (PA) of two ground-state atoms. The Franck-Condon overlap integrals between the harmonically trapped atom-pair states and the ultra-long-range molecular vibrational states are estimated and compared with their values for a pair of untrapped free atoms in the low-energy scattering state. We find that the binding between a pair of ground-state atoms by a harmonic trap has significant effect on the Franck-Condon integrals and thus can be used to influence PA. Trap-induced binding between two ground-state atoms may facilitate coherent PA dynamics between the two atoms and the photoassociated diatomic molecule.