论文标题
超速分子系统中范德华电位的精确溶液和量子缺陷理论
Exact Solutions and Quantum Defect Theory for van der Waals Potentials in Ultracold Molecular Systems
论文作者
论文摘要
在本文中，我们为2D和3DSchrödinger方程提供了精确的两体解，并具有各向同性范德华的势\（\ pm 1/r^6 \）。基于这些解决方案，我们开发了适用于准2D和3D几何形状的分析量子缺陷理论（QDT），并将其应用于限制在这些几何形状中的超声层极性分子的散射特性和结合状态。有趣的是，我们发现在有吸引力的（排斥性的）范德华电势情况下，短距离相互作用可以通过无限的方屏障（有限平方井）有效地建模，这会导致量子缺陷参数中的狭窄而密集（宽且稀疏）的谐振结构。在准2D有吸引力的情况下，形状共振可以在不同部分波浪中以有序的方式出现，其特征是随着散射能的变化而急剧跳跃。此外，从QDT得出的低能分析扩展与确切的数值结果非常吻合，从而验证了我们分析方法在描述由长距离范德华相互作用控制的两体物理学时的准确性和实用性。
In this paper, we have provided exact two-body solutions to the 2D and 3D Schrödinger 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.
