 02334 Frederic FAURE, Stephane NONNENMACHER, Stephan DE BIEVRE
 Scarred eigenstates for quantum cat maps of minimal periods
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Jul 30, 02

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Abstract. 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.
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