Scarred eigenstates for quantum cat maps of minimal periods. (English) Zbl 1033.81024

Faure, Frédéric; Nonnenmacher, Stéphane; De Bièvre, Stephan

Commun. Math. Phys. 239, No. 3, 449-492 (2003).

Summary: In this paper we construct a sequence of eigenfunctions of the “quantum Arnold’s cat map” that, in the semiclassical limit, shows a strong scarring phenomenon on the periodic orbits of the dynamics. More precisely, those states have a semiclassical limit measure that is the sum of 1/2 the normalized Lebesgue measure on the torus plus 1/2 the normalized Dirac measure concentrated on any a priori given periodic orbit of the dynamics. It is known (the Schnirelman theorem) that “most” sequences of eigenfunctions equidistribute on the torus. The sequences we construct therefore provide an example of an exception to this general rule. Our method of construction and proof exploits the existence of special values of \(\hbar\) for which the quantum period of the map is relatively “short”, and a sharp control on the evolution of coherent states up to this time scale. We also provide a pointwise description of these states in phase space, which uncovers their “hyperbolic” structure in the vicinity of the fixed points and yields more precise localization estimates.

MSC:

81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q20	Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis

Keywords:

semiclassical limit; Schnirelman theorem