Strong Scarring of Logarithmic Quasimodes

TL;DR

This paper constructs specific semiclassical quasimodes that strongly concentrate on a hyperbolic periodic orbit on a compact surface, demonstrating persistent localization despite their energy width shrinking with the semiclassical parameter.

Contribution

It explicitly constructs quasimodes with logarithmic energy width that exhibit strong scarring on hyperbolic orbits, extending previous results to more general settings.

Findings

Constructed quasimodes with energy width of order epsilon * h / |log h|

Demonstrated persistent localization (scarring) on hyperbolic orbits

Generalized previous results to broader classes of operators on surfaces.

Abstract

We consider a semiclassical (pseudo)differential operator on a compact surface $(M,g)$, such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit $\gamma$ at some energy $E_0$. For any $\epsilon>0$, we then explicitly construct families of quasimodes of this operator, satisfying an energy width of order $\epsilon \frac{h}{|\log h|}$ in the semiclassical limit, but which still exhibit a "strong scar" on the orbit $\gamma$, i.e. that these states have a positive weight in any microlocal neighbourhood of $\gamma$. We pay attention to optimizing the constants involved in the estimates. This result generalizes a recent result of Brooks \cite{Br13} in the case of hyperbolic surfaces. Our construction, inspired by the works of Vergini et al. in the physics literature, relies on controlling the propagation of Gaussian wavepackets up to the Ehrenfest time.