Isomorphisms between Algebras of Semiclassical Pseudodifferential Operators

TL;DR

This paper classifies isomorphisms between algebras of semiclassical pseudodifferential operators, showing they are realized by conjugation with specific microlocal operators linked to symplectic transformations.

Contribution

It extends the classification of algebra isomorphisms to the semiclassical setting, identifying their structure via microlocal elliptic operators and symplectic transformations.

Findings

Isomorphisms are given by conjugation with microlocal elliptic operators and $h$-FIOs.

Classifies algebra isomorphisms in the semiclassical pseudodifferential context.

Connects algebra isomorphisms to symplectic geometry via microlocal analysis.

Abstract

Following the work of Duistermaat-Singer \cite{DS} on isomorphisms of algebras of global pseudodifferential operators, we classify isomorphisms of algebras of microlocally defined semiclassical pseudodifferential operators. Specifically, we show that any such isomorphism is given by conjugation by $A = BF$, where $B$ is a microlocally elliptic semiclassical pseudodifferential operator, and $F$ is a microlocal $h$-FIO associated to the graph of a local symplectic transformation.