Low energy limit for the resolvent of some fibered boundary operators

TL;DR

This paper characterizes the low energy behavior of the resolvent of certain fibered boundary Dirac operators using pseudodifferential calculus, extending previous results and providing explicit proofs of Fredholm properties.

Contribution

It offers a pseudodifferential characterization of the low energy limit of the resolvent for fibered boundary Dirac operators, generalizing prior work and including explicit Fredholm property proofs.

Findings

Pseudodifferential characterization of the low energy limit of the resolvent.

Extension of Guillarmou and Sher's results to fibered boundary operators.

Explicit proof that the Dirac operator is Fredholm on weighted Sobolev spaces.

Abstract

For certain Dirac operators $\eth_{\phi}$ associated to a fibered boundary metric $g_{\phi}$, we provide a pseudodifferential characterization of the limiting behavior of $(\eth_{\phi}+k\gamma)^{-1}$ as $k\searrow 0$, where $\gamma$ is a self-adjoint operator anti-commuting with $\eth_{\phi}$ and whose square is the identity. This yields in particular a pseudodifferential characterization of the low energy limit of the resolvent of $\eth_{\phi}^2$, generalizing a result of Guillarmou and Sher about the low energy limit of the resolvent of the Hodge Laplacian of an asymptotically conical metric. As an application, we use our result to give a pseudodifferential characterization of the inverse of some suspended version of the operator $\eth_{\phi}$. One important ingredient in the proof of our main theorem is that the Dirac operator $\eth_{\phi}$ is Fredholm when acting on suitable weighted Sobolev spaces. This result has been known to experts for some time and we take this as an occasion to provide a complete explicit proof.