Subprincipal Controlled Quasimodes and Spectral Instability

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.