Solvability and limit bicharacteristics

TL;DR

This paper investigates the solvability of certain pseudodifferential operators with real principal symbols, focusing on the behavior of bicharacteristics near points where the principal symbol vanishes to second order, and establishes conditions for non-solvability.

Contribution

It introduces a detailed analysis of limit bicharacteristics for non-principal type operators and provides criteria for non-solvability based on the behavior of the subprincipal symbol.

Findings

Limit bicharacteristics are smooth curves obtained as limits of bicharacteristics.

Uniform bounds on curvature and linearization are established along tangent Lagrangian manifolds.

Non-solvability occurs when the imaginary part of the subprincipal symbol's ratio with the Hamilton vector field's norm switches sign and becomes unbounded.

Abstract

We shall study the solvability of pseudodifferential operators which are not of principal type. The operator will have real principal symbol and we shall consider the limits of bicharacteristics at the set where the principal symbol vanishes of at least second order. The convergence shall be as smooth curves, then the limit bicharacteristic is a smooth curve. We shall also need uniform bounds on the curvature of the characteristics at the bicharacteristics, but only along the tangents of a given Lagrangean manifold. This gives uniform bounds on the linearization of the normalized Hamilton flow on the tangent space of this manifold at the bicharacteristics. If the quotient of the imaginary part of the subprincipal symbol with the norm of the Hamilton vector field switches sign from $-$ to $+$ on the bicharacteristics and becomes unbounded as they converge to the limit, then the operator is not solvable at the limit bicharacteristic.