  In a recent experiment [Matsuda et al, Science 370, 1324 (2020)], a quasi
two-dimensional (2D), long-lived and strongly interacting diatomic polar
molecular gas was successfully prepared via controllable electric field
technique. Surprisingly, the effective repulsive and attractive Van der Waals
interactions of two molecules would emerge when scanning the strength of the
electric fields. Those results were also generalized to the three-dimensional
(3D) case in a later experiment [J. Li et al, Nature Physics 17, 1144 (2021)].
Motivated by these experiments, in this paper we provide the two-body exact
solutions for the 2D and 3D Schr\"{o}dinger equation with isotropic Van der
Waals potentials ($\pm1/r^{6}$). Furthermore, base on these exact solutions, we
build the analytical quantum defect theory (QDT) for quasi-2D and 3D
geometries, and then apply QDT to study the scattering properties and bound
states of two ultracold polar molecules confined in quasi-2D and 3D geometries.
Interestingly, we find that for the attractive (repulsive) Van der Waals
potential cases, the two-body short range potential can be approximated by an
square barrier with infinity height (square potential with finite depth) which
yields the wide (narrow) and dense (dilute) resonances of the quantum defect
parameter. For the quasi-2D attractive case, the scattering resonances of
different partial waves can orderly happen which is featured by the phase jumps
when varying the scattering energy. The analytical expansions in the low energy
limit shows a consistent agreement to the numerical results.
