Abstract

This note proves that the annihilation operator of a quantum harmonic oscillator admits an invariant distributionally -scrambled linear manifold for any . This is a positive answer to Question 1 by Wu and Chen (2013).


A dynamical system is a pair , where is a complete metric space without isolated points and the map is continuous. Throughout this paper, let and .

Sharkovskii’s amazing discovery [1], as well as Li and Yorke’s famous work which introduced the concept of “chaos” 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 “divergent pairs” , which are proximal but not asymptotic, in the sense that where denotes the th iteration of .

A generalization of the concept of Li-Yorke chaos is distributional chaos, introduced by Schweizer and Smítal [3] in 1994.

Let be a dynamical system. For any pair of points and any , let where denotes the cardinality of set . Define lower and upper distributional functions generated by , , and , as respectively. A dynamical system is said to be distributionally -chaotic for a given if there exists an uncountable subset such that for any pair of distinct points , one has for all and . The set is a distributionally -scrambled set and the pair a distributionally -chaotic pair. If is distributionally -chaotic for any given , then is said to exhibit maximal distributional chaos.

The quantum harmonic oscillator is the quantum-mechanical 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 a harmonic potential at the vicinity of a stable equilibrium point. Transmutation of the quantum harmonic oscillator may be described by the (time-dependent) Schrödinger equation as with a wave function , displacement , mass , frequency , and Planck constant . The nondimensionalized steady states in terms of eigenfunctions in the separable Hilbert space form an orthonormal basis: where is the th Hermite polynomial. The natural phase space for the quantum harmonic oscillator is the Schwartz class, also called Schwartz space of rapidly decreasing functions in , defined as Here, is an infinite-dimensional Fréchet space with topology defined by the system of seminorms of the form This topology on can be equivalently introduced by the metric It follows directly from (9) that . For any and any , denote . The quantum harmonic oscillator may be equivalently described in terms of the annihilation operator and its adjoint . According to the basic properties of Hermite polynomials, one has given by Meanwhile, it is not difficult to check that for any and any , where . The acts as a kind of backward shift on the space . In fact, it is a special weighted backward shift on the Fréchet space (see [68] for the recent results on this topic).

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

Question. Is there an invariant manifold of such that is a distributionally -scrambled set for any ?
This paper gives a positive answer to the question above; see the following theorem.

Theorem 1. There exists an invariant manifold such that is a distributionally -scrambled set under for any .

Proof. Let , , and for . Arrange all odd prime numbers by the natural order “” and denote them by . For any , set Take a point such that Since if , we have that, for any , This implies that, for any , Hence .
Take , where . Clearly, is an invariant linear manifold under . Given two fixed points with , according to the construction of , there exist , such that and .
Now, we assert that is a distributionally -chaotic pair for any .
First, observe that for any , and . Combining this with (14), it follows that for any fixed and any , as . Meanwhile, it is easy to see that, for any ,
For any fixed , one can choose a such that . It is clear that, for any , This, together with (16) and (17), leads to that there exists a such that for any and any ,
This implies that, for any , Consequently,
Second, since , there exists such that . It follows from (11) that, for any , For any fixed , there exists such that . According to the choice of , it follows that for all , . This implies that Then, for any , Combining this with the fact that the function is increasing, it follows that, for any , Hence, for any ,
Summing up the above discussions, since both and are arbitrary, it follows that is an invariant distributionally -scrambled linear manifold for any .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was completed during the author’s visit to the City University of Hong Kong. This paper was supported by YBXSZC20131046, the Scientific Research Fund of Sichuan Provincial Education Department (no. 14ZB0007).