Symmetry, Integrability and Geometry: Methods and Applications Pseudo-Bosons from Landau Levels ⋆ SIGMA 6 (2010), 093, 9 pages

We construct examples of pseudo-bosons in two dimensions arising from the Hamiltonian for the Landau levels. We also prove a no-go result showing that non-linear combinations of bosonic creation and annihilation operators cannot give rise to pseudo-bosons.


Introduction
In a series of recent papers [1,2,3,4,5,6], we have investigated some mathematical aspects of the so-called pseudo-bosons, originally introduced by Trifonov 1 in [8]. They arise from the canonical commutation relation [a, a † ] = 1 1 upon replacing a † by another (unbounded) operator b not (in general) related to a: [a, b] = 1 1. We have shown that, under suitable assumptions, N = ba and N † = a † b † can be both diagonalized, and that their spectra coincide with the set of natural numbers (including 0), N 0 . However the sets of related eigenvectors are not orthonormal (o.n.) bases but, nevertheless, they are automatically biorthonormal. In most of the examples considered so far, they are bases of the Hilbert space of the system, H, and, in some cases, they turn out to be Riesz bases.
In [9] and [10] some physical examples arising from quantum mechanics have been discussed. In particular, these examples have suggested the introduction of a difference between what we have called regular pseudo-bosons and pseudo-bosons, to better focus on what we believe are the mathematical or on the physical aspects of these particles. Indeed all the examples of regular pseudo-bosons considered so far arise from Riesz bases [4], with a rather mathematical construction, while pseudo-bosons are those which one can find when starting with the Hamiltonian of some realistic quantum system.
In this paper, after a short review of the general framework, we discuss a two-dimensional example arising from the Hamiltonian of the Landau levels. It should be stressed that this example is of a completely different kind than those considered in [10], where a modified version of the Landau levels have been considered.
We close the paper with a no-go result, suggesting that non-linear combinations of ordinary bosonic creation and annihilation operators, even if they produce pseudo-bosonic commutation rules, cannot satisfy the Assumptions of our construction, see Section 2.

The commutation rules
In this section we will review a d-dimensional version of what originally proposed in [1,6].
Let H be a given Hilbert space with scalar product ·, · and related norm · . We introduce d pairs of operators, a j and b j , j = 1, 2, . . . , d, acting on H and satisfying the following commutation rules j, k = 1, 2, . . . , d. Of course, these collapse to the CCR's for d independent modes if b j = a † j , j = 1, 2, . . . , d. It is well known that a j and b j are unbounded operators, so they cannot be defined on all of H. Following [1], and writing D ∞ (X) := ∩ p≥0 D(X p ) (the common domain of all the powers of the operator X), we consider the following: There exists a non-zero ϕ 0 ∈ H such that a j ϕ 0 = 0, j = 1, 2, . . . , d, and

Assumption 2. There exists a non-zero
. Under these assumptions we can introduce the following vectors in H: . . for all j = 1, 2, . . . , d. Let us now define the unbounded operators N j := b j a j and N j := N † j = a † j b † j , j = 1, 2, . . . , d. Each ϕ n belongs to the domain of N j , D(N j ), and Ψ n ∈ D(N j ), for all possible n. Moreover, N j ϕ n = n j ϕ n , N j Ψ n = n j Ψ n .
Under the above assumptions, and if we chose the normalization of Ψ 0 and ϕ 0 in such a way that Ψ 0 , ϕ 0 = 1, we find that This means that the sets F Ψ = {Ψ n } and F ϕ = {ϕ n } are biorthonormal and, because of this, the vectors of each set are linearly independent. If we now call D ϕ and D Ψ respectively the linear span of F ϕ and F Ψ , and H ϕ and H Ψ their closures, then This means, in particular, that both F ϕ and F Ψ are bases of H. The resolution of the identity in the bra-ket formalism looks like n |ϕ n Ψ n | = n |Ψ n ϕ n | = 1 1.
Let us now introduce the operators S ϕ and S Ψ via their action respectively on F Ψ and F ϕ : for all n, which also imply that Ψ n = (S Ψ S ϕ )Ψ n and ϕ n = (S ϕ S Ψ )ϕ n , for all n. Hence In other words, both S Ψ and S ϕ are invertible and one is the inverse of the other. Furthermore, we can also check that they are both positive, well defined and symmetric [1]. Moreover, it is possible to write these operators as These expressions are only formal, at this stage, since the series may not converge in the uniform topology and the operators S ϕ and S Ψ could be unbounded. Indeed we know [11], that two biorthonormal bases are related by a bounded operator, with bounded inverse, if and only if they are Riesz bases 2 . This is why in [1] we have also considerered Assumption 4. F ϕ and F Ψ are both Riesz bases.
Therefore, as already stated, S ϕ and S Ψ are bounded operators and their domains can be taken to be all of H. While Assumptions 1, 2 and 3 are quite often satisfied, [12], it is quite difficult to find physical examples satisfying also Assumption 4. On the other hand, it is rather easy to find mathematical examples satisfying all the assumptions, see [1,6]. This is why in [9] we have introduced a difference in the notation: we have called pseudo-bosons (PB) those satisfying the first three assumptions, while, if they also satisfy Assumption 4, they are called regular pseudo-bosons (RPB).
As already discussed in our previous papers, these d-dimensional pseudo-bosons give rise to interesting intertwining relations among non self-adjoint operators, see in particular [3] and references therein. For instance, it is easy to check that This is related to the fact that the spectra of, say, N 1 and N 1 , coincide and that their eigenvectors are related by the operators S ϕ and S Ψ , in agreement with the literature on intertwining operators [13,14].

The example
In this section we will consider an example arising from a quantum mechanical system, i.e. a single electron moving on a two-dimensional plane and subject to a uniform magnetic field along the z-direction. Taking = m = eB c = 1, the Hamiltonian of the electron is given by the operator where we have used minimal coupling and the symmetric gauge A = 1 2 (−y, x, 0). The Hilbert space of the system is H = L 2 (R 2 ).
The spectrum of this Hamiltonian is easily obtained by first introducing the new variables In terms of P 1 and Q 1 the single electron Hamiltonian, H 1 , can be rewritten as The transformation (3) is part of a canonical map from the variables (x, y, p x , p y ) to (Q 1 , Q 2 , P 1 , P 2 ), where which can be used to construct a second Hamiltonian The two Hamiltonians correspond to two opposite magnetic fields, respectively along +k and −k. Let us now introduce the operators k = 1, 2, together with their adjoints. Then [A k , A † l ] = δ k,l 1 1, the other commutators being zero. In terms of these operators we can write H k = A † k A k + 1 2 1 1, k = 1, 2, whose eigenvectors are Φ n , for k = 1, 2. It is natural to introduce the sets F k := Φ (k) n , n ≥ 0 , k = 1, 2, and the closures of their linear span, H 1 and H 2 . Hence, by construction, F k is an o.n. basis of H k . Moreover, we can also introduce an o.n. basis of H as the set F Φ whose vectors are defined as follows:

Pseudo-bosons in H 1
Let us now define the following operators: Hence, we recover (1) for d = 1 in H 1 . We want to show that A 1 (α) and B 1 (α) generate PB in H 1 which are not regular.
To begin with, we define ϕ    (B 1 (α)). This follows from the fact that, since
Before considering Assumption 2, it is convenient to observe that, introducing the following invertible and densely defined operator U 1 (α) := e αA 2 1 , we can write for all n ≥ 0. Of course, ϕ (1) n (α) is well defined for all n ≥ 0 since, as we have seen, B 1 (α) n ϕ (1) 0 (α) is well defined for all complex α. Now, if we define (at least formally, at this stage) it is possible to show that, if |α| < 1 2 : (i) Ψ . It is furthermore possible to check that, for the same values of α, Let us prove point (iii) above. We have, for all n ≥ 0, which converges inside the disk |α| < 1 2 . In particular, if n = 0, this implies the statement in (i) above. The proof of (ii) is trivial and the last equality in (6) can be deduced using (4) and (5) The proof of Assumption 3 goes as follows: First of all, as we have already stated, it is possible to check that for all n ≥ 0 we have ϕ k , for some constants {d k , k = 0, 1, . . . , n − 1}. Secondly, using induction on n and this simple remark we can prove that, if f ∈ H 1 is such that f, ϕ  k 's, whose set is complete in H 1 . Hence f = 0, so that F ϕ (1) is also complete in H 1 .
As a consequence, being the vectors of F ϕ (1) linearly independent and complete in H 1 , they are a basis of H 1 . In particular we find that, for all f ∈ H 1 , the following expansion holds true: Then, for all f, g ∈ H 1 , which, since f could be any vector in H 1 , implies that g = ∞ k=0 ϕ (1) n (α): F Ψ (1) is a basis of H 1 as well, and Assumption 3 is satisfied. Finally, Assumption 4 is not satisfied since, for instance, the operator U 1 (α) † −1 is unbounded [11]. Remark 1. It might be worth stressing that, while it is quite easy to check that the set F ϕ (1) is complete in D(U (α) † ), it is not trivial at all to check that it is also complete in H 1 . This is the reason why we have used the above procedure.

Pseudo-bosons in H 2
In this subsection we will consider an analogous construction in H 2 , i.e. in the Hilbert space related to the uniform magnetic field along −k. To make the situation more interesting, and to avoid repeating essentially the same procedure considered above, instead of introducing an operator like e βA 2 2 we consider with β ∈ C. Then we define These are pseudo-bosonic operators in H 2 : [A 2 (β), B 2 (β)] = 1 1, and A 2 (β) † = B 2 (β), for β = 0. Then, once again, it may be interesting to consider Assumptions 1-4. If |β| < 1 2 Assumption 1 is satisfied: let us define (formally, for the moment) ϕ 0 , which implies in particular that Of course we have now to check that ϕ (2) n (β) is a well defined vector of H 2 for all n ≥ 0. This would make the above formal definition rigorous. The computation of U 2 (β)Φ (2) n follows the same steps as that for U 1 (α) † −1 Φ (1) n of the previous section, and we get the same conclusion: the power series obtained for U 2 (β)Φ (2) n 2 converges if |β| < 1 2 , so that Φ (2) n ∈ D(U 2 (β)) for all n ≥ 0, inside this disk.
As for Assumption 2, this is also satisfied: to prove this it is enough to take Ψ n , which is clearly a vector in H 2 since it is a finite linear combination of Φ (2) 0 , Φ 1 , . . . , Φ (2) n . This means that the vectors Ψ (2) n (β) := n are well defined in H 2 for all n, independently of β. Once again we deduce that the vectors constructed here are biorthonormal, and that they are eigenstates of two operators which are the adjoint one of the other, and which are related to H 2 by a similarity transformation: which in coordinate representation looks like In particular we find that for all n ≥ 0. The same arguments used previously prove that F ϕ (2) := {ϕ (2) n (β), n ≥ 0} and F Ψ (2) := {Ψ (2) n (β), n ≥ 0} are both complete in H 2 . More than this: they are biorthonormal bases but not Riesz bases.