An Isomorphism Theorem for Some Linear Elliptic Differential Operators on Quasi-Asymptotically Conical Manifolds

TL;DR

This paper establishes an isomorphism theorem for certain linear elliptic operators on quasi-asymptotically conical manifolds, extending understanding of their analytical properties in complex geometric settings.

Contribution

It introduces an isomorphism theorem for elliptic operators on quasi-asymptotically conical manifolds with unbounded potentials, advancing analysis on such geometric structures.

Findings

Proves an isomorphism theorem for the operator $\mathcal{P}$

Characterizes the spectral properties of $\mathcal{P}$ on these manifolds

Extends elliptic theory to manifolds with polyhomogeneous metrics and unbounded potentials

Abstract

We study a linear elliptic differential operator of the form $\mathcal{P}=\Delta + V - \lambda$ on a quasi-asymptotically conical manifold $(M, g)$, where $g$ is a polyhomogeneous metric and $V$ is a $b$-vector field that is unbounded with respect to the metric $g$.