Invariant Distributionally Scrambled Manifolds for an Annihilation ...

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May 15, 2014 - This note proves that the annihilation operator of a quantum harmonic oscillator admits an invariant distributionally -scrambled.
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 754960, 4 pages http://dx.doi.org/10.1155/2014/754960

Research Article Invariant Distributionally Scrambled Manifolds for an Annihilation Operator Xinxing Wu School of Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China Correspondence should be addressed to Xinxing Wu; [email protected] Received 20 January 2014; Accepted 15 May 2014; Published 27 May 2014 Academic Editor: Alfredo Peris Copyright Β© 2014 Xinxing Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This note proves that the annihilation operator of a quantum harmonic oscillator admits an invariant distributionally πœ€-scrambled linear manifold for any 0 < πœ€ < 2. This is a positive answer to Question 1 by Wu and Chen (2013).

A dynamical system is a pair (𝑋, 𝑓), where 𝑋 is a complete metric space without isolated points and the map 𝑓 : 𝑋 β†’ 𝑋 is continuous. Throughout this paper, let N = {1, 2, 3, . . .} and Z+ = {0, 1, 2, . . .}. Sharkovskii’s amazing discovery [1], as well as Li and Yorke’s famous work which introduced the concept of β€œchaos” known as the Li-Yorke chaos today in a mathematically rigorous way [2], has activated sustained interest and provoked the rapid advancement of discrete chaos theory in the last decades. Since then, several other rigorous definitions of chaos have been proposed. Each of their definitions tries to describe one kind of unpredictability in the evolution of the system dynamics. This was also the original idea of Li and Yorke. In Li and Yorke’s study [2], they suggested considering β€œdivergent pairs” (π‘₯, 𝑦), which are proximal but not asymptotic, in the sense that lim inf 𝑑 (𝑓𝑛 (π‘₯) , 𝑓𝑛 (𝑦)) = 0, π‘›β†’βˆž

lim sup𝑑 (𝑓𝑛 (π‘₯) , 𝑓𝑛 (𝑦)) > 0,

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π‘›β†’βˆž

where 𝑓𝑛 denotes the 𝑛th iteration of 𝑓. A generalization of the concept of Li-Yorke chaos is distributional chaos, introduced by Schweizer and SmΒ΄Δ±tal [3] in 1994.

Let (𝑋, 𝑓) be a dynamical system. For any pair of points π‘₯, 𝑦 ∈ 𝑋 and any 𝑛 ∈ N, let 󡄨 󡄨 𝐹 (π‘₯, 𝑦, 𝑑, 𝑛) = 󡄨󡄨󡄨󡄨{𝑗 ∈ N : 𝑑 (𝑓𝑗 (π‘₯) , 𝑓𝑗 (𝑦)) < 𝑑, 1 ≀ 𝑗 ≀ 𝑛}󡄨󡄨󡄨󡄨 , (2) where |𝐴| denotes the cardinality of set 𝐴. Define lower and upper distributional functions R β†’ [0, 1] generated by 𝑓, π‘₯, and 𝑦, as 1 𝐹π‘₯,𝑦 (𝑑, 𝑓) = lim inf 𝐹 (π‘₯, 𝑦, 𝑑, 𝑛) , π‘›β†’βˆž 𝑛

(3)

1 βˆ— (𝑑, 𝑓) = lim sup 𝐹 (π‘₯, 𝑦, 𝑑, 𝑛) , 𝐹π‘₯,𝑦 π‘›β†’βˆž 𝑛

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respectively. A dynamical system (𝑋, 𝑓) is said to be distributionally πœ€-chaotic for a given πœ€ > 0 if there exists an uncountable subset 𝐷 βŠ‚ 𝑋 such that for any pair of distinct βˆ— (𝑑, 𝑓) = 1 for all 𝑑 > 0 and points π‘₯, 𝑦 ∈ 𝐷, one has 𝐹π‘₯,𝑦 𝐹π‘₯,𝑦 (πœ€, 𝑓) = 0. The set 𝐷 is a distributionally πœ€-scrambled set and the pair (π‘₯, 𝑦) a distributionally πœ€-chaotic pair. If (𝑋, 𝑓) is distributionally πœ€-chaotic for any given 0 < πœ€ < diam 𝑋, then (𝑋, 𝑓) is said to exhibit maximal distributional chaos. The quantum harmonic oscillator is the quantummechanical analog of the classical harmonic oscillator. It is one of the most important models in quantum mechanics [4, 5], because an arbitrary potential can be approximated by

2

Abstract and Applied Analysis

a harmonic potential at the vicinity of a stable equilibrium point. Transmutation of the quantum harmonic oscillator may be described by the (time-dependent) SchrΒ¨odinger equation as βˆ’

πœ•πœ“ β„Ž2 πœ•2 πœ“ π‘šπœ”2 2 + π‘₯ πœ“ = π‘–β„Ž , 2 2π‘š πœ•π‘₯ 2 πœ•π‘‘

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with a wave function πœ“(π‘₯, 𝑑), displacement π‘₯, mass π‘š, frequency πœ”, and Planck constant β„Ž. The nondimensionalized steady states in terms of eigenfunctions in the separable Hilbert space H = 𝐿2 (βˆ’βˆž, +∞) form an orthonormal basis:

Applying a result of Godefroy and Shapiro [9], Gulisashvili and MacCluer [10] proved that the annihilation operator π‘ŽΜ‚ is Devaney chaotic. Then, Duan et al. [11] obtained that π‘ŽΜ‚ is also Li-Yorke chaotic. However, it follows directly from [12, Theorem 4.1] that this holds trivially. Oprocha [13] showed that π‘ŽΜ‚ is distributionally πœ€-chaotic with πœ€ = 1/16. Recently, in [14] it was further shown that π‘ŽΜ‚ exhibits distributional πœ€chaos for any 0 < πœ€ < 2 and that the principal measure of π‘ŽΜ‚ is 1. Moreover, Wu and Chen [15] proved that π‘ŽΜ‚ admits an invariant distributionally πœ€-scrambled set for any 0 < πœ€ < 2 and posed the following question.

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Question. Is there an invariant manifold 𝐷 of Ξ¦ such that 𝐷 is a distributionally πœ€-scrambled set for any 0 < πœ€ < 2? This paper gives a positive answer to the question above; see the following theorem.

where 𝐻𝑛 (π‘₯) = (βˆ’1)𝑛 𝑒π‘₯ (d𝑛 /dπ‘₯𝑛 )π‘’βˆ’π‘₯ is the 𝑛th Hermite polynomial. The natural phase space for the quantum harmonic oscillator is the Schwartz class, also called Schwartz space Ξ¦ of rapidly decreasing functions in H, defined as

Theorem 1. There exists an invariant manifold 𝐸 βŠ‚ Ξ¦ such that 𝐸 is a distributionally πœ€-scrambled set under π‘ŽΜ‚ for any 0 < πœ€ < 2 = diam Ξ¦.

2

πœ“π‘› (π‘₯) =

π‘’βˆ’π‘₯ /2 𝐻𝑛 (π‘₯) βˆšβˆšπœ‹2𝑛 𝑛!

,

𝑛 = 0, 1, . . . ,

2

2

∞

∞

𝑛=0

𝑛=0

󡄨 󡄨2 Ξ¦ = {πœ™ ∈ H : πœ™ = βˆ‘ 𝑐𝑛 πœ“π‘› , βˆ‘ 󡄨󡄨󡄨𝑐𝑛 󡄨󡄨󡄨 (𝑛 + 1)π‘Ÿ < +∞, βˆ€π‘Ÿ > 0} . (7) Here, Ξ¦ is an infinite-dimensional FrΒ΄echet space with topology defined by the system of seminorms π‘π‘Ÿ (β‹…) of the form ∞

∞

1/2

󡄨 󡄨2 π‘π‘Ÿ (πœ™) = π‘π‘Ÿ ( βˆ‘ 𝑐𝑛 πœ“π‘› ) = ( βˆ‘ 󡄨󡄨󡄨𝑐𝑛 󡄨󡄨󡄨 (𝑛 + 1)π‘Ÿ ) 𝑛=0

.

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Proof. Let 𝐿 1 = L1 = 2, 𝐿 𝑛 = 2𝐿 1 +β‹…β‹…β‹…+𝐿 π‘›βˆ’1 , and L𝑛 = βˆ‘π‘›π‘—=1 𝐿 𝑗 for 𝑛 > 1. Arrange all odd prime numbers by the natural order β€œ