Strong scarring of logarithmic quasimodes

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 γ at some energy E 0 . For any ε > 0, we then explicitly construct families of quasimodes of this operator, satisfying an energy width of order ε | log | in the semiclassical limit, but which still exhibit a "strong scar" on the orbit γ, i.e. that these states have a positive weight in any microlocal neighbourhood of γ. We pay attention to optimizing the constants involved in the estimates. This result generalizes a recent result of Brooks [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.

Scarring of quasimodes on hyperbolic manifolds

“…The bulk of the paper is devoted to establishing (1). The reference [13,Sec. 6] shows how to deduce (2) from (1) using spectral projection; see Lemma 7.7 below.…”

“…It is important to add that the construction of quasimodes which localize along closed geodesics, or more generally along smooth invariant submanifolds, has a long and rich history. For a brief exposition of this history, see the introduction of [13] and the references therein.…”

“…Ψ will concentrate on M exactly when ψ will concentrate on the origin of the x-plane. This improves on the construction in [13] where the ϕ were particular localized wave packets on M of width Cγ | log | so that their evolution (especially self-interference) was non-trivial and needed to be dealt with using a particular normal form due to Sjöstrand [25]; we will discuss this in greater detail shortly. The use of exact eigenfunctions, or more generally quasimodes of specified width, therefore simplifies the construction.…”

“…Expansion around the fixed point. The following result is inspired by the work of Combescure-Robert [11] and its proof is an r-dimensional version of Proposition 4.9 in [13]. Hence, we only provide the necessary details and leave the remaining elements to the reader.…”

“…To get a narrower quasimode we use the usual time-averaging proceudre originally due to Vergini-Schneider [27] and also employed in the work [13].…”