Semiclassical pseudodifferential operators and resolvent parametrices on manifolds with ends

TL;DR

This paper develops a new class of semiclassical pseudodifferential operators on manifolds with ends, enabling the construction of resolvent parametrices and proving essential self-adjointness of elliptic operators.

Contribution

It introduces a metric-independent pseudodifferential calculus on manifolds with ends, applicable to asymptotically conical and hyperbolic geometries.

Findings

Constructed resolvent parametrices for elliptic operators on manifolds with ends.

Proved essential self-adjointness of elliptic symmetric differential operators.

Developed a pseudodifferential calculus independent of Riemannian metric.

Abstract

We construct a parametrix of a resolvent of elliptic differential operators acting on half-densities on manifolds with ends. The construction is carried out by introducing suitable pseudodifferential operators compatible with the end structure. Our class of pseudodifferential operators and symbols is independent of the choice of Riemannian metric on the manifold and applicable to both of asymptotically conical and hyperbolic manifolds. As an application, we prove the essential self-adjointness of elliptic symmetric differential operators on manifolds.