Construction of quasimodes for non-selfadjoint operators via propagation of Hagedorn wave-packets

TL;DR

This paper develops a method to construct quasimodes for non-selfadjoint semiclassical operators by propagating Hagedorn wave-packets, focusing on boundary pseudo-spectrum points and invariant structures.

Contribution

It introduces a novel propagation technique for Hagedorn wave-packets to build quasimodes concentrating on non-damped phase-space points in non-selfadjoint operators.

Findings

Constructed quasimodes at the boundary of the pseudo-spectrum.

Applied method to perturbations of the harmonic oscillator.

Demonstrated concentration on non-damped periodic orbits or tori.

Abstract

We construct quasimodes for some non-selfadjoint semiclassical operators at the boundary of the pseudo-spectrum using propagation of Hagedorn wave-packets. Assuming that the imaginary part of the principal symbol of the operator is non-negative and vanishes on certain points of the phase-space satisfying a subelliptic finite-type condition, we construct quasimodes that concentrate on these non-damped points. More generally, we apply this technique to construct quasimodes for non-selfadjoint semiclassical perturbations of the harmonic oscillator that concentrate on non-damped periodic orbits or invariant tori satisfying a weak-geometric-control condition