Subprincipal Control of Pseudospectral Quasimodes, II

TL;DR

This paper extends the analysis of semiclassical subprincipal controlled quasimodes for pseudodifferential operators with double characteristics, focusing on tangential intersections of bicharacteristics and operator factorization effects.

Contribution

It advances the understanding of quasimodes in complex characteristic scenarios, including tangential intersections and factorable operators, building on previous work with transversal intersections.

Findings

Identified cases where subprincipal control influences pseudospectrum stability.

Developed normal form microlocal analysis for operators with double characteristics.

Explored the impact of operator factorization on quasimode behavior.

Abstract

In this paper, we continue the analysis of the effects of semiclassical sub principal controlled quasimodes, approximate solutions to P(h)u(h,b), depending on the subprincipal symbol b, which can give spectral insta bility (pseudospectrum). We consider a pseudodifferential operator, which has double zeros for the principal symbol, p. This means that p = dp = 0 in a small neighborhood. In the first paper in this series, we considered operators with transversal inter sections of bicharacteristics. Now we study operators with tangential in tersections of bicharacteristics, as well as with double characteristics for p. We put the pseudodifferential operator on normal form microlocally, and use a model operator, P(h) to test for quasimodes. We demonstrate two cases where this happens. We shall also continue with more advanced cases, when the operators are factorable to P(h) = P2(h)P1(h,B), thus annihilating the subprincipal control over the quasimodes.