Abstract

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 shift 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 (in H.O. units) is shown to be given by \(\Delta =-2\frac{\delta }{\pi }\) , also exact in the limit of infinite scattering length ( \(\delta =\pm \frac{\pi }{2}\) ) in which case \(\Delta =\mp 1\) . Numerical investigation shows that this expression otherwise differs 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 result 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 solid state physics referred to as Fumi’s theorem. 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 is in this context referred to as the phase-shift approximation a term also used throughout this report