Microlocal Projector for Complete Flow

TL;DR

This paper extends the microlocal projector construction to sublevel sets of autonomous functions with complete Hamiltonian flows, highlighting the optimality of the completeness condition through a counterexample.

Contribution

It generalizes microlocal projector construction to new settings and demonstrates the necessity of the completeness condition with a non-example.

Findings

Microlocal projector construction is extended to sublevel sets of autonomous functions.

Completeness of Hamiltonian flow is shown to be a necessary condition.

A non-example illustrates the failure of the projector property without completeness.

Abstract

In this note we generalize the construction of microlocal projector to the sublevel set of autonomous function with complete Hamiltonian flow under some mild conditions. Furthermore, we mention that the condition of being complete is optimal, by providing a non-example that violates the projector property.