On the invariance of maximal distributional chaos under an annihilation operator

Abstract

a b s t r a c tIt is known that certain physical systems, which do not generate deterministic chaos under conventional frameworks, may generate such complex behavior in a quantum mechanical setting. In this paper, it is proved that the annihilation operator of an unforced quantum harmonic oscillator admits an invariant distributionally ϵ-scrambled set for any 0 < ϵ < 2, showing that this operator can exhibit maximal distributional chaos on an uncountable invariant subset.

Invariance of chaos from backward shift on the Köthe sequence space

“…By employing methods developed in [3], we proved that for a bounded operator defined on a Banach space, Li-Yorke chaos, Li-Yorke sensitivity, spatiotemporal chaos and distributional chaos in a sequence are all equivalent, and they are all strictly stronger than sensitivity [29]. Further results of [18] were extended to maximal distributional chaos for the annihilation operator of a quantum harmonic oscillator in [27,33]. In 2009, Martínez-Giménez et al [16] provided sufficient conditions for uniform distributional chaos under backward shift.…”

Some Kinds of Distributional Chaos for Non-Autonomous Discrete Systems

“…The related concept distributional chaotic pair as two points for which the statistical distribution of distances between the orbits does not converge, and Schweizer and Smital (1994) proved that the existence of a single distributional chaotic pair is equivalent to the positive topological entropy (and some other notions of chaos) when restricted to the compact interval case. Since then, distributional chaos has been widely concerned in dynamical system theory (see Smítal and Štefánková, 2004;Balibrea et al, 2005;Martínez-Giménez et al, 2009;Liao et al, 2009;Oprocha, 2009;Li, 2011;Dvorakova, 2011;Wu and Chen, 2013;Shao et al, 2018). Smítal and Štefánková (2004) showed that the two notions of distributional chaos used in the paper, for continuous maps of a compact metric space, are invariants of topological conjugation.…”

“…has positive topological entropy and Y f displays distributional chaos of type 1, but not conversely. Next year, Li (2011) showed that for a continuous selfmap f of a compact metric space X and any integer 0 , showing that this operator can exhibit maximal distributional chaos on an uncountable invariant subset (in (Wu and Chen, 2013)).…”

Invariant Distributionally Scrambled Manifolds for an Annihilation Operator

“…Recently, in [14] it was further shown that̂exhibits distributionalchaos for any 0 < < 2 and that the principal measure of is 1. Moreover, Wu and Chen [15] proved that̂admits an invariant distributionally -scrambled set for any 0 < < 2 and posed the following question.…”