Analytical solutions of the Schrödinger equation for two confined atoms with van der Waals interaction

Abstract

We derive solutions of the Schrödinger equation for the isotropic van der Waals interaction in a symmetric harmonic trap, with the recent approach [arXiv:2207.09377 (2022)] to handle the multi-scale long-range potential. Asymptotic behaviors of these solutions are then obtained for $r\to 0$ and $r\to \infty$. We further deduce the energy spectrum of the two-body relative motion and relate the spectrum to scattering lengths for $s$ wave and $p$ wave. These results can be used to research trapped atom-atom collisions and energy spectra.

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