On the $L^p$ Spectrum of the Dirac operator

TL;DR

This paper extends the understanding of the $L^p$-spectra of Dirac operators on open manifolds, identifying conditions for spectrum independence of p and characterizing spectra for various geometric classes.

Contribution

It provides new sufficient conditions for the $L^p$-spectrum to be p-independent and generalizes classes of manifolds with maximal spectra for Dirac operators.

Findings

$L^p$-spectrum is p-independent under certain geometric conditions

The $L^p$-spectrum can be the entire real line for specific manifolds

Maximal $L^2$-spectrum on manifolds with nonnegative Ricci or asymptotically flat geometry

Abstract

Our main goal in the present paper is to expand the known class of open manifolds over which the $L^2$-spectrum of a general Dirac operator and its square is maximal. To achieve this, we first find sufficient conditions on the manifold so that the $L^p$-spectrum of the Dirac operator and its square is independent of $p$ for $p\geq 1$. Using the $L^1$-spectrum, which is simpler to compute, we generalize the class of manifolds over which the $L^p$-spectrum of the Dirac operator is the real line for all $p$. We also show that by applying the generalized Weyl criterion, we can find large classes of manifolds with asymptotically nonnegative Ricci curvature, or which are asymptotically flat, such that the $L^2$-spectrum of a general Dirac operator and its square is maximal.