Analysis on noncompact manifolds and Index Theory: Fredholm conditions and Pseudodifferential operators

We provide Fredholm conditions for compatible differential operators on certain Lie manifolds (that is, on certain possibly non-compact manifolds with nice ends). We discuss in more detail the case of manifolds with cylindrical, hyperbolic, and Euclidean ends, which are all covered by particular instances of our results. We also discuss applications to Schr\"odinger operators with singularities of the form r^{-2\gamma}$, $\gamma \in \RR_+$.