# On a Chaotic Weighted Shiftzpdp+1/dzp+1 of Order p in Bargmann Space

Authors: Abdelkader Intissar
Journal: Advances in Mathematical Physics
 (2011)

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## Abstract

This paper is devoted to the study of the chaotic properties of some specific backward
shift unbounded operatorsHp=A*pAP+1; p=0,1,… realized as differential operators in Bargmann space, where A and A* are the standard Bose annihilation and creation operators such that [A,A*]=I.


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## Body

## 1. Introduction

It is well known that linear operators in finite-dimensional linear spaces cannot be chaotic but the nonlinear operator may be. Only in infinite-dimensional linear spaces can linear operators have chaotic properties. These last properties are based on the phenomenon of hypercyclicity or the phenomen of nonwandercity.The study of the phenomenon of hypercyclicity originates in the papers by Birkoff  [1] and Maclane  [2] who show, respectively, that the operators of translation and differentiation, acting on the space of entire functions, are hypercyclic.The theories of hypercyclic operators and chaotic operators have been intensively developed for bounded linear operators; we refer to  [1, 3–5] and references therein. For a bounded operator, Ansari asserts in  [6] that powers of a hypercyclic bounded operator are also hypercyclic.For an unbounded operator, Salas exhibits in  [7] an unbounded hypercyclic operator whose square is not hypercyclic. The result of Salas shows that one must be careful in the formal manipulation of operators with restricted domains. For such operators, it is often more convenient to work with vectors rather than with operators themselves.Now, letT be an unbounded operator on a separable infinite dimensional Banach space X. A point ϕ is called wandering if there exists an open set U containing ϕ such that for some n0<∞ and for all n>n0 one has Tn(U)⋂U=∅. (In other words, the neighbourhood eventually never returns). A point ϕ is called nonwandering if it is not wandering.A closed subspaceE⊂X has hyperbolic structure if: E=Eu⊕Es, TEu=Eu, and TEs=Es, where Eu (the unstable subspace) and Es (the stable subspace) are closed. In addition, there exist constants τ(0<τ<1) and C>0, such that:(i) for anyΦ∈Eu,k∈ℕ,Cτ-k||Φ||≤||TkΦ||,(ii) for anyΨ∈Es,k∈ℕ,||TkΨ||≤Cτk||Ψ||.T is said to be a nonwandering operator relative to E which has hyperbolic structure if the set of periodic points of T is dense in E.For the nonwandering operators, they are new linear chaotic operators. They are relative to hypercyclic operators, but different from them in the sense that some hypercyclic operators are not non-wandering operators and there also exists a non-wandering operator, which does not belong to hypercyclic operators (see [8], Remark   3.5). In fact, suppose T is a bounded linear operator and T is invertible; if T is a hypercyclic operator, then σ(T)⋂∂D≠Φ (see [9], Remark   4.3) but if T is a non-wandering operator, then σ(T)⋂∂D=Φ where ∂D is the unit circle.Now, when a linear operator is not invertible, there exist operators which are not only non-wandering but also hypercyclic. Recently, these theories began to be developed on some concrete examples of unbounded linear operators; see  [1, 10–12]. On the basis of the work in  [13], we study the phenomenons of chaoticity of some specific backward shift unbounded operators Hp=A*pAp+1=zp(dp+1/dzp+1);p=0,1,… realized as differential operators in Bargmann space  [14] (the space of entire functions with Gaussian measure), where A and A* are the standard Boson annihilation and creation operators satisfying the commutation relation (1.1)[A,A*]=I.Of special interest is a representation of these operatorsA and A*  as linear operators in a separable Hilbert spanned by eigenvectors |n〉;n=0,1,… of the positive semidefinite number operator  N=A*A.One has the well-known relations(1.2)A|n〉=n|n-1〉,A*|n〉=n+1|n+1〉.
We denote the Bargmann space  [14] by (1.3)B={ϕ:C⟶Centire,∫C|ϕ(z)|2e-|z|2dxdy<∞}.The scalar product onB is defined by(1.4)〈ϕ,ψ〉=∫Cϕ(z)ψ(z)¯e-|z|2dxdy,
and the associated norm is denoted by ∥·∥.An orthonormal basis ofB is given by(1.5)en(z)=znn!;n=0,1,….B is closed in L2(ℂ,dμ(z)), where the measure dμ(z)=e-|z|2dxdy and is closed related to L2(ℝ) by an unitary transform of L2(ℝ) onto B given in [14] by the following integral transform:(1.6)ϕ(z)=∫Re-(1/2)(z2+u2)+2uzf(u)du.Iff∈L2(ℝ) the integral converges absolutely.In this Bargmann representation, the annihilator and creator operators are defined by(1.7)Aϕ(z)=ddzϕ(z)withdomainD(A)={ϕ∈B;ddzϕ∈B},A*ϕ(z)=zϕ(z)withdomainD(A*)={ϕ∈B;zϕ∈B}.Now, we define(1.8)Bp={ϕ∈B;djdzjϕ(0)=0,0≤j≤p}.An orthonormal basis ofBp is given by(1.9)en(z)=znn!;n=p+1,p+2,….Hence, a family of weighted shiftsHp is defined as follows:(1.10)Hp=A*pAp+1withdomainD(Hp)={ϕ∈B;Hpϕ∈B}⋂Bp.Remark 1.1.
(i) Forp=0, the operator H0=A is the derivation in Bargmann space, and it is the celebrated quantum annihilation operator.
(ii)H0*en=n+1en+1 is a weighted shift with weight ωn=n+1 for n=0,1,….
(iii) It is known thatH0 with its domain D(H0) is a chaotic operator in Bargmann space.
(iv)H0ϕλ(z)=λϕλ(z) for all λ∈ℂ where ϕλ(z)=∑n=0∞(λn/n!)en(z) and ∥ϕλ∥2=e|λ|2.
(v) The functione-|λ|2ϕλ(z) is called a coherent normalized quantum optics (see [15, 16]).Remark 1.2.
(i) Forp=1, the operator H1=A*A2=z(d2/dz2) has as adjoint the operator H1*=z2(d/dz).
(ii)H1*en=nn+1en+1 is a weighted shift with weight ωn=nn+1 for n=1,… and it is known that  H1+H1* is a not self-adjoint operator and is chaotic in Bargmann space [13]. This operator plays an essential role in Reggeon field theory (see [17, 18]).
(iii) The operatorsHp arising also in the Jaynes-Cummings interaction models, see for example a model introduced by Obada and Abd Al-Kader in [19], the interaction Hamiltonian for the model is
(1.11)HI=-∑p=0∞{Ω1eiϕ1e-(η12/2)(iη1)2p+1p!(p+1)!Hp*+Ω2eiϕ2e-(η22/2)(iη2)2p+1p!(p+1)!Hp}σ-+h⋅c,
where Ωj are the Rabi frequencies and and ηj2 are the Lamb-Dicke; j = 1,2.         The operators σ- and σ+ act on the ground state |g〉 and excited state |e〉 as follows:  σ±|g〉=(1±1/2)|e〉 and σ±|e〉=(1∓1/2)|g〉.
(iv) OnBp,p=0,1,…, which is the orthogonal of span {en;n≤p} in Bargmann space, the adjoint of Hp is Hp*=zp+1(dp/dzp) such that(1.12)Hp*en=ωnen+1withweightωn=n+1n!(n-p)!forn≥p≥0.
This paper is organized as follows:   in Section2, we recall the definition of the chaoticity for an unbounded operator following Devaney and sufficient conditions on hypercyclicity of unbounded operators given by Bés-Chan-Seubert theorem [10].  As our operator Hp is an unilateral weighted backward shift with an explicit weight, we use the results of Bés et al. to proof the chaoticity of Hp in Bargmann space (we can also use the results of Bermúdez et al. [11] to proof the chaoticity of our operator Hp). Then, we construct the hyperbolic structure associated to Hp. In the appendix, we present a direct proof of the chaoticity of Hp based on the Baire Category theorem. The last theorem is essential to proof that the operator is topologically transitive and can be used for interested reader.


## 2. Chaoticity of the OperatorHp=zp(dp+1/dzp+1) on Bp

Definition 2.1.
LetT be an unbounded linear operator on a separable infinite dimensional Banach X with domain D(T) dense in X and such that Tn is closed for all positive integers n.(a)
The operatorT is hypercyclic if there exists a vector f∈D(T) such that Tnf∈D(T) and if the orbit {f,Tf,T2f,…} is dense in X. The vector f is called a hypercyclic vector of T.(b)
A vectorg∈D(T) is called a periodic point of T if there exists m such that Tmg=g. The operators having both dense sets of periodic points and hypercyclic vectors are said to be chaotic following the definition of Devaney  [20, 21].
Sufficient conditions for the hypercyclicity of an unbounded operator are given in the following Bés-Chan-Seubert theorem:Theorem 2.2 (Bés-Chan-Seubert [10]).
LetX be a separable infinite dimensional Banach, and let T be a densely defined linear operator on X. Then, T is hypercyclic if(i)
Tm is a closed operator for all positive integers m,(ii)
there exists a dense subsetF of the domain D(T) of T and a (possibly nonlinear and discontinuous) mapping S:F→F so that TS is the identity on F and Tn,Sn→0 pointwise on F as n→∞.Theorem 2.3.
LetB be the Bargmann space with orthonormal basis en(z)=zn/n!.  Let Hp=zp(dp+1/dzp+1)=A*pAp+1 with domain D(Hp)={ϕ∈B;Hpϕ∈B}⋂Bp, where Bp={ϕ∈B;(dj/dzj)ϕ(0)=0,0≤j≤p}. Then, Hp is a chaotic operator.Remark 2.4.
(i) Following the ideas of Gross-Erdmann in [4, 5] or the Theorem  2.4 of Bermúdez et al. [11], we can use a test on the weight of Hp to give a proof of the chaoticity of Hp.
We choose to give a proof under lemma form based on the theorem of Bès et al. recalled above, we also indicate in the appendix the utilization of the Baire category theorem in the hepercyclicity theory and we prove thatHp possesses a certain "sensitivity to initial conditions" though this property is redundant in Devaney's definition (see Banks et al. in [20]).
(ii) LetT be an unbounded operator on separable infinite dimensional Banach X. It may happen that vector f ∈D(T), but Tf fails to be in the domain of T. We can exhibit a closed operator whose square is not. For example, the operator acting on L2(0,1)×L2(0,1) defined by T(u,v)(x)=(v′(x),f(x)v(0)) with domain D(T)=L2(0,1)×H1(0,1), where v′(x) is the derivative of v(x) and f is a function in H1(0,1) with f(0)=1, where H1(0,1) is the classical Sobolev space. Then T, is a closed operator and D(T2)=D(T), where D(T2) is the domain of T2 but the operator T2 is not closed and has not closed extension.This operator can, for example, justify the asumption (a) of the Definition 2.1 for the unbounded linear operators.Lemma 2.5.
For each positive integerm, the operator (Hp)m, with domain D((Hp)m)={ϕ∈B;(Hp)mϕ∈B}⋂Bp, is a closed operator.Proof.
As(Hp)m is closed if and only if the graph G((Hp)m) is a closed linear manifold of Bp×Bp, then let (ϕn,(Hp)mϕn) be a sequence in G((Hp)m) which converges to (ϕ,ψ) in Bp×Bp. As ϕn converges to ϕ in Bp, then zp(dp+1/dzp+1)ϕn converges to zp(dp+1/dzp+1)ϕ pointwise on ℂand (Hp)mϕn converges to (Hp)mϕ pointwise on ℂ. As (Hp)mϕn converges to ψ, we deduce that (Hp)m)ϕ=ψ and ϕ∈D((Hp)m), hence G((Hp)m) is closed.Lemma 2.6.
LetHp=zp(dp+1/dzp+1)  with domain D(Hp)={ϕ∈B;Hpϕ∈B}⋂Bp, where Hpen=ωn-1en-1, en(z)=zn/n!, and ωn=n+1(n!/(n-p)!) for n≥p≥0. Then,  Hp is hypercyclic.Proof.
LetF={ϕk(z)=∑n=pkanen(z)}. This space is dense in Bp.
LetSp: F→F and Spen=(1/ωn)en+1;n≥p≥0.
Then,HpSpϕk(z)=ϕk(z), that is, HpSp=I|F.
Now,  as[Hp]ken=0 for all k>n≥p we deduce that any element of F can be annihilated by a finite power kn of Hp since as [∏j=nkn+nωj]-1→0 when kn→∞, we have(2.1)Spknen=[∏j=nkn+nωj]-1ek+n⟶0inBp.
Now, the hypercyclicity of Hp follows from the theorem of Bés et al. recalled above.Lemma 2.7.
LetHp=zp(dp+1/dzp+1)  with domain D(Hp)={ϕ∈B;Hpϕ∈B}⋂Bp, where Hpen=ωn-1en-1, en(z)=(zn/n!), and ωn=n+1(n!/(n-p)!) for n≥p≥0. Then, there exist k>0 and g∈D(Hpk) such that Hpkg(z)=g(z).Proof.
Letλ∈ℂ and
(2.2)gλ(z)=ep(z)+∑n=p+1∞λn-pωpωp+1⋯ωn-1en(z),
then gλ is in the domain of Hp and it is an eigenvector for Hp corresponding to eigenvalue λ, therefore it is a periodic point of Hp. λ is a root of unity.
In fact, letr>0 and |λ|<r, then as
(2.3)lim⁡⁡∏j=pn-1ωj=∞whenn⟶∞,
there exist n0>0 and q<1 such that
(2.4)r(ωpωp+1⋯ωn-1)(1/n)≤qforn≥n0,
since for |λ|<r, we have
(2.5)|λ|(n-p)(ωpωp+1⋯ωn-1)2≤q2n;n≥n0
and gλ is in Bargmann space. Now as,
(2.6)〈gλ,ep〉=1,〈gλ,en+1〉=λn-p+1ωpωp+1⋯ωn,
we get
(2.7)|〈gλ,en+1〉|2=λ2(n-p+1)(ωpωp+1⋯ωn)2,|〈gλ,en+1〉|2(ωn)2=λ2(n-p+1)(ωpωp+1⋯ωn-1)2≤q2n|λ|2forn≥n0.
We deduce that(2.8)∑n=p∞|〈gλ,en+1〉|2(ωn)2<∞,
that is,gλ∈D(Hp).
Thus,  we get(2.9)Hpgλ=Hpep+∑n=p+1∞λn-pωpωp+1⋯ωn-1⋅ωn-1en-1=λ∑m=p∞λm-pωpωp+1⋯ωm-1em=λ{ep+∑m=p+1∞λm-pωpωp+1⋯ωm-1em}=λgλ.
Therefore,gλ is the eigenvector corresponding to the eigenvalue λ and gλ is a periodic point of Hp, where λ is a root of unity.Lemma 2.8.
The set of periodic points ofHp is dense in Bp.Proof.
Let(2.10)G=span⁡{gλ;λisarootofunity}.G is dense in Bp, otherwise there exists nonzero vector g∈Bp which is orthogonal to G.
Let(2.11)g(z)=∑n=p∞bnen(z)suchthat〈g,gλ〉=0,foreachgλ∈G,f(λ)=〈g,gλ〉for|λ|<1,f(λ)=0for|λ|=1.
f(λ) is a continuous function on the closed unit disc which is holomorphic on the interior and vanishes at each root of unity, hence on the entire unit circle, hence f(λ) vanishes for all |λ|≤1. We deduce that bn=0 for n≥p, then G is dense in Bp.Remark 2.9.
(i) The Lemmas2.5, 2.6, and 2.8 show the chaoticity of Hp.
(ii) The Theorem2.3 generalizes the result of [12] on the annihilation operator in Bargmann space.Definition 2.10.
LetT be an unbounded linear operator on a separable infinite dimensional Banach X whose domain D(T) is dense in X, and let Tn be closed for all positive integers n.(a)
A closed subspaceE⊂X has hyperbolic structure if E=Eu⊕Es, TEu=Eu, and TEs=Es, where Eu (the unstable subspace) and Es (the stable subspace) are closed. In addition, there exist constants τ(0<τ<1) and C>0, such that:
(i)
For anyΦ∈Eu,k∈ℕ,Cτ-k||Φ||≤||TkΦ|| (the vectors of Eu are exponentially expanded, we say they belong to the unstable subspace Eu).(ii)
For anyΨ∈Es,k∈ℕ,||TkΨ||≤Cτk||Ψ|| (some vectors are contracted exponentially fast by the iterates of the operator T, we say they belong to the stable subspace Es).(b)
If there exists a closed subspaceE⊂X which has hyperbolic structure relative to T and the set of periodic points of T is dense in E, then T is said to be a nonwandering  operator relative to E following the definition of Tian et al. [8].
SinceHp is chaotic operator on Bp, so Hp has dense set of periodic points on Bp, we only need to construct an hyperbolic structure associated to it in Bp to obtain:Theorem 2.11.
LetB be the Bargmann space with orthonormal basis en(z)=zn/n!.
LetHp=zp(dp+1/dzp+1)=A*pAp+1 with domain D(Hp)={ϕ∈B;Hpϕ∈B}⋂Bp, where Bp={ϕ∈B;(dj/dzj)ϕ(0)=0,0≤j≤p}.
Then,Hp  is a nonwandering operator.Proof.
We construct a closed invariant subspaceE⊂Bp such that E has hyperbolic structure.
Forλ∈ℂ, the function defined by (2.2).
gλ(z)=ep(z)+∑n=p+1∞(λn-p/ωpωp+1⋯ωn-1)en(z) is in the domain of Hp and is an eigenvector for Hp corresponding to the eigenvalue λ.
LetEu=span{gλ;|λ|>1}¯, Es=span{gλ;0<|λ|<1}¯, and E=Eu⊕Es, where ⊕ represents direct sum.
We will verify thatE has an hyperbolic structure.
Forϕ∈Eu, there exists a sequence (ai),i=1,2,… such that(2.12)ϕ(z)=∑i=1∞aigλi(z)=∑i=1∞ai∑n=p∞λin-pωpωp+1⋯ωn-1en(z).
And for each positive integer m, we have(2.13)‖(Hp)mϕ‖=‖(Hp)m∑i=1∞aigλi‖=‖(Hp)m∑i=1∞aiλim∑n=p∞λin-pωpωp+1⋯ωn-1en‖≥μm‖ϕ‖,
whereμ=min{|λi|;|λi|>1}.
Next, we will proveEu is the invariant subspace of Hp.
Letϕ∈Eu, then(2.14)ϕ(z)=∑i=1∞aigλi(z)=∑i=1∞ai∑n=p∞λin-pωpωp+1⋯ωn-1en(z)=Hp∑i=1∞bigλi(z),
wherebi=ai/λithenEu⊂HpEu.
Now forψ∈HpEu, then there exists ϕ∈Eu, such that ψ=HpΦ=∑i=1∞cigλi, where ci=λiai. Therefore, HpEu⊂Eu.
Similarly, letEs=span{gλ;0<|λ|<1}¯, we deduce that HpEs=Es and if we chose τ=1/μ, then we have (2.15)‖(Hp)mϕ‖≥‖ϕ‖τm.
Then, E has hyperbolic structure and Hp is nonwandering operator relative to E.
Here, the linear spaceEs is formed by the (spectral) subspace corresponding to the eigenvalues of Hp of modulus less than 1, while the unstable subspace Eu corresponds to those of modulus greater than 1.As in  [22], we can use the Gazeau-Klauder formalism to construct the coherent states of this operator Hp and investigate some properties of these coherent states (see [23]).We conclude that main results of this work can be considered in  [24] as an introduction to study of the operators Hp,m=zp(∂p+m/dzp+m) with p=0,1,… and m=1,2,… particularly, to study the chaoticity of Hp,m+Hp,m*.



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