Analytical Solutions of the Schrödinger Equation for Two Confined Particles with the van der Waals Interaction

We derive exact analytical solutions to the Schrödinger equation featuring a dual-scale potential, namely, a blend of a van der Waals (vdW) potential and an isotropic harmonic potential. The asymptotic behaviors of these solutions as \(r\rightarrow 0\) and \(r\rightarrow \infty \) are also elucidated. These results are obtained through the approach we recently developed [arXiv: 2207.09377]. Using our results, we further calculate the s-wave and p-wave energy spectrums of two particles confined in an isotropic harmonic trap, with vdW inter-particle interaction. We compare our exact results and the ones given by the zero-range pseudopotential (ZRP) approaches, with either energy-dependent or energy-independent s-wave scattering length \(a_s\) or p-wave scattering volume \(V_p\). It is shown that the results of ZRP approaches with energy-dependent \(a_s\) or \(V_p\) consist well with our exact ones, when the length scale \(\beta _6\) of the vdW potential equals to or less than the length scale \(a_h\) of the confinement potential. Furthermore, when \(\beta _6\gg a_h\) (e.g., \(\beta _6=10a_h\)) all the ZRP approaches fail. Our results are helpful for the research of confined ultracold atoms or molecules with strong vdW interactions.