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.

受 Landau–Ginzburg 模型的启发，我们研究了具有有界几何的非紧流形上的 Witten 变形，以及莫尔斯函数f接近无穷大的增长的一些温顺条件。我们证明作用于 平滑 $L^2$ 复形的Witten变形 $d_{Tf}$ 的上同调与f的Thom-Smale复形的上同调以及某对的相对上同调同构 $(M, U)$ 对于足够大的T . 我们为威滕拉普拉斯算子的特征形式建立了 Agmon 估计，它在通过威滕的瞬子复形识别这些上同调方面起着至关重要的作用，根据小特征值的威滕拉普拉斯算子的特征空间定义。作为应用，我们在此设置中获得了强莫尔斯不等式。