Motivated by the Landau–Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function f near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of the Thom–Smale complex of f as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large T. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten’s instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application, we obtain the strong Morse inequalities in this setting.

Asymptotic expansions of heat kernels and heat traces of Schrödinger operators on non-compact spaces are rarely explored, and even for cases as simple as $${\mathbb {C}}^n$$ 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.

Asymptotic expansions of heat kernels and heat traces of Schrödinger operators on non-compact spaces are rarely explored, and even for cases as simple as 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.