Subprincipal Controlled Quasimodes and Spectral Instability

TL;DR

This paper investigates semiclassical quasimodes associated with pseudodifferential operators having double multiplicity in their principal symbol, focusing on how the subprincipal symbol influences spectral instability and pseudospectrum, with implications for operator factorization.

Contribution

It introduces a novel analysis of subprincipal-controlled quasimodes in operators with bicharacteristic intersections, and explores conditions for spectral stability and operator factorization.

Findings

Sign changes in the imaginary part of the subprincipal symbol lead to localized quasimodes.

Factorization of the operator can eliminate the influence of the subprincipal symbol.

The study extends to more complex operators with tangential bicharacteristic intersections.

Abstract

Here we explore, in a series of articles, semiclassical quasimodes u(h,b), approximative solutions P(h)u(h,b)\sim 0, depending on $0<h<1$, and on b, the subprincipal symbol. We study a pseudodifferential operator with transversal intersections of bicharacteristics, where the principal symbol has double multiplicity, $p=dp=0$, in a small neigborhood $\Omega$. Because of this fact, we instead study the subprincipal symbol b, and we can conclude that we get transport equations depending on b where sign changes for the imaginary part of b give approximative solutions with small support. These modes are used to estimate spectral instability, or the pseudospectrum. We also investigate the possibility that we can factorize the model operator as $P(h)=h^2P_1P_2,$ in this way actually annihilating the subprincipal symbol, thus there is no condition for the imaginary part of b. In a follow-up article, we examine different cases for more complex operators with tangential intersections of bicharacteristics, thereby generalizing the findings here.