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 → 0 and r → ∞ 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 β 6 of the vdW potential equals to or less than the length scale a h of the confinement potential. Furthermore, when β 6 ≫ a h (e.g., β 6 = 10 a h ) all the ZRP approaches fail. Our results are helpful for the research of confined ultracold atoms or molecules with strong vdW interactions.