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A novel modified oscillator method of calculating energy eigenvalues for colliding ultracold atoms tightly confined in harmonic potentials is presented and applied to trapped spin-polarized metastable helium atoms. The perturbed harmonic oscillator problem is characterized by a long asymptotic region beyond the effective range of the interatomic potential, and a very efficient method for integrating inwards through this outer region is developed. The calculated eigenvalues for states with $l=0$ agree closely with those computed directly from the radial Schr\"{o}dinger equation for the trapped atoms using a discrete variable method, and with those computed self-consistently from an energy-dependent effective scattering length. Finally, the self-consistent method is generalized to collisions with $l \neq 0$ and applied to d-wave collisions of metastable helium atoms

Ultracold atomic collisions in tight harmonic traps: Quantum-defect model and application to metastable helium atoms