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

TL;DR

This paper establishes Fredholm conditions for differential operators on specific noncompact Lie manifolds with various ends, and explores applications to Schrödinger operators with singular potentials.

Contribution

It provides new Fredholm criteria for differential operators on noncompact manifolds with ends, extending index theory to these settings.

Findings

Fredholm conditions for differential operators on Lie manifolds with cylindrical, hyperbolic, and Euclidean ends

Application of results to Schrödinger operators with inverse power singularities

Analysis of index theory in noncompact geometric contexts

Abstract

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}$, $\gamma \in \RR_+$.