2002/07/31 by F. Faure, Fréderic Faure, S. Nonnenmacher +3 · 165 citations
Mathematics · Physics and Astronomy · #Eigenfunction #Eigenvalues and eigenvectors #Geometry #Lebesgue integration #Lebesgue measure #Limit (mathematics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Measure (data warehouse) #Nuclear physics research studies #Phase space #Physics #Pointwise #Pure mathematics #Quantum #Quantum chaos and dynamical systems #Quantum mechanics #Semiclassical physics #Torus #math-ph #math.MP #nlin.CD
paper · pdf · doi:10.1007/s00220-003-0888-3
published in Communications in Mathematical Physics 239(3), 449-492 (Springer Science+Business Media) · LaTeX, 49 pages, includes 10 figures. I added section 6.6. To be published in Commun. Math. Phys
arxiv created 2003/06/06 · openalex publication_date 2003/08/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper we construct a sequence of eigenfunctions of the ``quantum Arnold's cat map'' that, in the semiclassical limit, show 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 Planck's constant 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.