Index theory of the de Rham complex on manifolds with periodic ends

Abstract

We study the de Rham complex on a smooth manifold with a periodic end modeled on an infinite cyclic cover $\tilde{X}\to X$. The completion of this complex in exponentially weighted $L^2$ norms is Fredholm for all but finitely many exceptional weights determined by the eigenvalues of the covering translation map $H_{\ast }\left(\tilde{X}\right)\to H_{\ast }\left(\tilde{X}\right)$. We calculate the index of this weighted de Rham complex for all weights away from the exceptional ones.