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Abstract
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.