  We analyze a system of two colliding ultracold atoms under strong harmonic
confinement from the viewpoint of quantum defect theory and formulate a
generalized self-consistent method for determining the allowed energies. We
also present two highly efficient computational methods for determining the
bound state energies and eigenfunctions of such systems. The perturbed harmonic
oscillator problem is characterized by a long asymptotic region beyond the
effective range of the interatomic potential. The first method, which is based
on quantum defect theory and is an adaptation of a technique developed by one
of the authors (GP) for highly excited states in a modified Coulomb potential,
is very efficient for integrating through this outer region. The second method
is a direct numerical solution of the radial Schr\"{o}dinger equation using a
discrete variable representation of the kinetic energy operator and a scaled
radial coordinate grid. The methods are applied to the case of trapped
spin-polarized metastable helium atoms. The calculated eigenvalues agree very
closely for the two methods, and with those computed self-consistently using
the generalized self-consistent method.
