  The energy-spectrum of two point-like particles interacting in a 3-D
isotropic Harmonic Oscillator (H.O.) trap is related to the free scattering
phase-shifts $\delta$ of the particles by a formula first published by Busch et
al. It is here used to find an expression for the \it shift \rm of the energy
levels, caused by the interaction, rather than the perturbed spectrum itself.
In the limit of high energy (large quantum number $n$ of the H.O.) this shift
is shown to be given by $-2\frac{\delta}{\pi}$, also valid in the limit of
infinite as well as zero scattering length at all H.O. energies.
  Numerical investigation shows that the shifts differ from the exact result of
Busch et al, by less than $<\frac{1}{2}\%$ except for $n=0$ when it can be as
large as $\approx 2.5\%$.
  This approximation for the energy-shift is well known from another exactly
solvable model, namely that of two particles interacting in a spherical
infinite square-well trap (or box) of radius $R$ in the limit $R\rightarrow
\infty$, and/or in the limit of large energy. It is in this context referred to
as the \it phase-shift approximation \rm.
  It can be (and has been) used in (infinite) nuclear matter calculations to
calculate the two-body effective interaction in situations where in-medium
effects can be neglected. It has also been used in expressing the energy of
free electrons in a metal.
