Classification of order preserving isomorphisms between algebras of semiclassical operators

Following the work of Duistermaat and Singer on isomorphisms of algebras of global pseudodifferential operators, we classify order preserving 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 microlocally unitary $h$-FIO associated to the graph of a local symplectic transformation.