  We consider the heat equation associated to Schr\"{o}dinger operators acting
on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel
$L^p - L^q$ decay estimates are established, allowing the Schr\"{o}dinger
operator to have a non-trivial $L^2$-kernel. We also prove new decay estimates
for spatial derivatives of arbitrary order, in a general geometric setting. Our
main motivation is the application to stability of non-linear geometric
equations, primarily Ricci flow, which will be presented in a companion paper.
The arguments in this paper use that many geometric Schr\"{o}dinger operators
can be written as the square of Dirac type operators. By a remarkable result of
Wang, this is even true for the Lichnerowicz Laplacian, under the assumption of
a parallel spinor. Our analysis is based on a novel combination of the Fredholm
theory for Dirac type operators on ALE manifolds and recent advances in the
study of the heat kernel on non-compact manifolds.
