Symbol calculus for Gevrey pseudodifferential operators and adiabatic projectors

TL;DR

This paper develops a symbol calculus for Gevrey pseudodifferential operators, constructs a parametrix for elliptic operators, and derives exponential estimates for adiabatic projectors within this framework.

Contribution

It introduces a new family of norms for formal Gevrey symbols that form a Banach algebra, enabling the construction of parametrices and exponential estimates.

Findings

Constructed a parametrix for elliptic Gevrey pseudodifferential operators.

Established a Banach algebra structure for Gevrey symbol norms.

Derived exponential estimates for adiabatic projectors in the Gevrey setting.

Abstract

We construct a parametrix of an elliptic Gevrey pseudodifferential operator, by introducing a family of norms for formal Gevrey symbols with the property of a Banach algebra under the symbol calculus. As an application, we obtain exponential estimates for adiabatic projectors in the Gevrey setting.