Geometric results for hyperbolic operators with spectral transition of the Hamilton map

TL;DR

This paper investigates a class of non-effectively hyperbolic operators with spectral transitions of the Hamilton map, providing normal forms, Hamiltonian analysis, and a factorization to facilitate microlocal energy estimates.

Contribution

It introduces a novel analysis of hyperbolic operators with spectral transitions, including normal symplectic coordinates and a factorization approach.

Findings

Normal symplectic coordinates are constructed.

Hamilton system analysis is performed.

A factorization result is established for microlocal energy estimates.

Abstract

In this paper we study a class of non-effectively hyperbolic operators vanishing of order 2 on a manifold, on a sub-region of which the spectral structure of the Hamilton map changes type. Suitable normal symplectic coordinates are found together with an analysis of the Hamilton system associated to the principal symbol and a factorization result, preparing the operator for a microlocal energy estimate, is finally proven.