On a Chaotic Weighted Shift z^pd^p+1/dz^p+1 of Order p in Bargmann Space

Remark 1.1. (i) For p = 0, the operator H₀ = A is the derivation in Bargmann space, and it is the celebrated quantum annihilation operator.

(ii) $$ is a weighted shift with weight $$ for n = 0,1, ….

(iii) It is known that H₀ with its domain D(H₀) is a chaotic operator in Bargmann space.

(iv) H₀ϕ_λ(z) = λϕ_λ(z) for all λ ∈ ℂ where $$ and $$.

(v) The function $$ is called a coherent normalized quantum optics (see [15, 16]).

Remark 1.2. (i) For p = 1, the operator H₁ = A^*A² = z(d²/dz²) has as adjoint the operator $$.

(ii) $$ is a weighted shift with weight $$ for n = 1, … and it is known that $$ 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 operators H_p 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

(iv) On B_p, p = 0,1, …, which is the orthogonal of span {e_n; n ≤ p} in Bargmann space, the adjoint of H_p is $$ such that

This paper is organized as follows:   in Section 2, 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 H_p is an unilateral weighted backward shift with an explicit weight, we use the results of Bés et al. to proof the chaoticity of H_p in Bargmann space (we can also use the results of Bermúdez et al. [11] to proof the chaoticity of our operator H_p). Then, we construct the hyperbolic structure associated to H_p. In the appendix, we present a direct proof of the chaoticity of H_p 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.

Definition 2.1. Let T 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 operator T 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 vector g ∈ D(T) is called a periodic point of T if there exists m such that T^mg = 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]). Let X be a separable infinite dimensional Banach, and let T be a densely defined linear operator on X. Then, T is hypercyclic if

• (i)

  T^m is a closed operator for all positive integers m,

• (ii)

  there exists a dense subset F 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. Let B be the Bargmann space with orthonormal basis $$. Let $$ with domain D(H_p) = {ϕ ∈ B; H_pϕ ∈ B} ⋂ B_p, where B_p = {ϕ ∈ B; (d^j/dz^j)ϕ(0) = 0,   0 ≤ j ≤ p}. Then, H_p 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 H_p to give a proof of the chaoticity of H_p.

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 that H_p possesses a certain "sensitivity to initial conditions" though this property is redundant in Devaney′s definition (see Banks et al. in [20]).

(ii) Let T 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 integer m, the operator $$, with domain $$, is a closed operator.

Proof. As $$ is closed if and only if the graph $$ is a closed linear manifold of B_p × B_p, then let $$ be a sequence in $$ which converges to (ϕ, ψ) in B_p × B_p. As ϕ_n converges to ϕ in B_p, then z^p(d^p+1/dz^p+1)ϕ_n converges to z^p(d^p+1/dz^p+1)ϕ   pointwise on ℂ  and $$ converges to $$ pointwise on ℂ. As $$ converges to ψ, we deduce that (H_p) ^m)ϕ = ψ and ϕ ∈ D((H_p) ^m), hence G((H_p) ^m) is closed.

Lemma 2.6. Let H_p = z^p(d^p+1/dz^p+1) with domain D(H_p) = {ϕ ∈ B; H_pϕ ∈ B}⋂B_p, where H_pe_n = ω_n−1e_n−1, $$, and $$ for n ≥ p ≥ 0. Then, H_p is hypercyclic.

Proof. Let $$. This space is dense in B_p.

Let S_p: F → F and S_pe_n = (1/ω_n)e_n+1; n ≥ p ≥ 0.

Then, H_pS_pϕ_k(z) = ϕ_k(z), that is, H_pS_p = I_|F.

Now, as [H_p] ^ke_n = 0 for all k > n ≥ p we deduce that any element of F can be annihilated by a finite power k_n of H_p since as $$ when k_n → ∞, we have

Lemma 2.7. Let H_p = z^p(d^p+1/dz^p+1) with domain D(H_p) = {ϕ ∈ B; H_pϕ ∈ B}⋂B_p, where H_pe_n = ω_n−1e_n−1, $$, and $$ for n ≥ p ≥ 0. Then, there exist k > 0 and $$ such that $$.

Proof. Let λ ∈ ℂ and

In fact, let r > 0 and |λ | < r, then as

that is, g_λ ∈ D(H_p).

Thus, we get

Therefore, g_λ is the eigenvector corresponding to the eigenvalue λ and g_λ is a periodic point of H_p, where λ is a root of unity.

Lemma 2.8. The set of periodic points of H_p is dense in B_p.

Proof. Let

Let

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 b_n = 0 for n ≥ p, then G is dense in B_p.

Remark 2.9. (i) The Lemmas 2.5, 2.6, and 2.8 show the chaoticity of H_p.

(ii) The Theorem 2.3 generalizes the result of [12] on the annihilation operator in Bargmann space.

Definition 2.10. Let T 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 subspace E⊂X has hyperbolic structure if E = E^u ⊕ E^s, TE^u = E^u, and TE^s = E^s, where E^u (the unstable subspace) and E^s (the stable subspace) are closed. In addition, there exist constants τ(0 < τ < 1) and C > 0, such that:

  • (i)

    For any Φ ∈ E^u,     k ∈ ℕ,     Cτ^−k||Φ|| ≤ ||T^kΦ|| (the vectors of E^u are exponentially expanded, we say they belong to the unstable subspace E^u).

  • (ii)

    For any Ψ ∈ E^s,     k ∈ ℕ,     ||T^kΨ||   ≤ Cτ^k| | Ψ|| (some vectors are contracted exponentially fast by the iterates of the operator T, we say they belong to the stable subspace E^s).

• (b)

  If there exists a closed subspace E⊂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].

Since H_p is chaotic operator on B_p, so H_p has dense set of periodic points on B_p, we only need to construct an hyperbolic structure associated to it in B_p to obtain:

Theorem 2.11. Let B be the Bargmann space with orthonormal basis $$.

Let $$ with domain D(H_p) = {ϕ ∈ B; H_pϕ ∈ B}⋂B_p, where B_p = {ϕ ∈ B; (d^j/dz^j)ϕ(0) = 0,   0 ≤ j ≤ p}.

Then, H_p is a nonwandering operator.

Proof. We construct a closed invariant subspace E ⊂ B_p such that E has hyperbolic structure.

For λ ∈ ℂ, the function defined by (2.2).

$$ is in the domain of H_p and is an eigenvector for H_p corresponding to the eigenvalue λ.

Let $$, $$, and E = E^u ⊕ E^s, where ⊕ represents direct sum.

We will verify that E has an hyperbolic structure.

For ϕ ∈ E^u, there exists a sequence (a_i),     i = 1,2, … such that

where    μ = min {|λ_i|; |λ_i| > 1}.

Next, we will prove E^u is the invariant subspace of H_p.

Let ϕ ∈ E^u, then

Now for $$, then there exists   ϕ ∈ E^u, such that $$, where c_i = λ_ia_i. Therefore, H_pE^u ⊂ E^u.

Similarly, let $$, we deduce that H_pE^s = E^s and if we chose τ = 1/μ, then we have

Here, the linear space E^s is formed by the (spectral) subspace corresponding to the eigenvalues of H_p of modulus less than 1, while the unstable subspace E^u corresponds to those of modulus greater than 1.