Exact Solutions and Quantum Defect Theory for van der Waals Potentials in Ultracold Molecular Systems

TL;DR

This paper derives exact solutions for van der Waals potentials in ultracold molecules, develops an analytical quantum defect theory for different geometries, and reveals how short-range interactions influence resonance structures and scattering properties.

Contribution

It provides the first exact solutions and an analytical QDT framework for 2D and 3D van der Waals potentials in ultracold molecular systems.

Findings

Exact solutions for (r) (r) (r) potentials

Analytical QDT accurately predicts scattering and bound states

Resonance structures depend on short-range interaction modeling

Abstract

In this paper, we have provided exact two-body solutions to the 2D and 3D Schr\"odinger equations with isotropic van der Waals potentials of the form \(\pm 1/r^6\). Based on these solutions, we developed an analytical quantum defect theory (QDT) applicable to both quasi-2D and 3D geometries, and applied it to study the scattering properties and bound-state spectra of ultracold polar molecules confined in these geometries. Interestingly, we find that in the attractive (repulsive) van der Waals potential case, the short-range interaction can be effectively modeled by an infinite square barrier (finite square well), which leads to narrow and dense (broad and sparse) resonance structures in the quantum defect parameter. In the quasi-2D attractive case, shape resonances can appear in an ordered fashion across different partial waves, characterized by sharp phase jumps as the scattering energy is varied. Furthermore, the low-energy analytical expansions derived from QDT show excellent agreement with the exact numerical results, validating the accuracy and usefulness of our analytical approach in describing two-body physics governed by long-range van der Waals interactions.