  Asymptotic expansions of heat kernels and heat traces of Schr\"odinger
operators on non-compact spaces are rarely explored, and even for cases as
simple as $\mathbb{C}^n$ with (quasi-homogeneous) polynomials potentials, it's
already very complicated. Motivated by path integral formulation of the heat
kernel, we introduced a parabolic distance, which also appeared in Li-Yau's
famous work on parabolic Harnack estimate. With the help of the parabolic
distance, we derive a pointwise asymptotic expansion of the heat kernel for the
Witten Laplacian with strong remainder estimate. When the deformation parameter
of Witten deformation and time parameter are coupled, we derive an asymptotic
expansion of trace of heat kernel for small-time $t$, and obtain a local index
theorem. This is the second of our papers in understanding Landau-Ginzburg
B-models on nontrivial spaces, and in subsequent work, we will develop the
Ray-Singer torsion for Witten deformation in the non-compact setting.
