Harmonic Oscillator Trap and the Phase-Shift Approximation

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.