  As is well-known, the Witten deformation of the De Rham complex computes the
De Rham cohomology. In this paper we study the Witten deformation on a
noncompact manifold and restrict it to differential forms which behave
polynomially near infinity. Such polynomial differential forms naturally appear
on manifolds with a cylindrical structure. We prove that the cohomology of the
Witten deformation acting on the complex of the polynomially growing forms can
be computed as the relative cohomology of the manifold with respect to the
negative remote fiber of the function. We show that the assumptions of our main
theorem are satisfied in a number of interesting special cases, including
generic real polynomials.
