Modified Landau levels, damped harmonic oscillator and two-dimensional pseudo-bosons

Modified Landau levels, damped harmonic oscillator and two-dimensional pseudo-bosons S. Twareque Ali F. Bagarello Jean Pierre Gazeau 3 Abstract In a series of recent papers one of us has analyzed in some details a class of elementary excitations called pseudo-bosons. They arise from a special deformation of the canonical commutation relation [a, a ] =, which is replaced by [a, b] =, with b not necessarily equal to a. Here, after a two-dimensional extension of the general framework, we apply the theory to a generalized version of the two-dimensional Hamiltonian describing Landau levels. Moreover, for this system, we discuss coherent states and we deduce a resolution of the identity. We also consider a different class of examples arising from a classical system, i.e. a damped harmonic oscillator. Department of Mathematics and Statistics, Concordia University, Montréal, Québec, CANADA H3G M8 e-mail: stali@mathstat.concordia.ca Dipartimento di Metodi e Modelli Matematici, Facoltà di Ingegneria, Università di Palermo, I-908 Palermo, ITALY e-mail: bagarell@unipa.it Home page: www.unipa.it/ bagarell 3 Laboratoire APC, Université Paris 7-Denis Diderot, 0, rue A. Domon et L. Duquet, 7505 Paris Cedex 3, France

I Introduction In a series of recent papers [,, 3, 4], one of us FB) has investigated some mathematical aspects of the so-called pseudo-bosons, originally introduced by Trifonov in [5]. They arise from the canonical commutation relation [a, a ] = upon replacing a by another unbounded) operator b not in general) related to a: [a, b] =. We have shown that 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 biorthogonal. In all 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. To our knowledge, not many physical consequences of this construction have been discussed up to now. For this reason, extending what two of us STA and FB) have already done in [6], we will construct here a two-dimensional model which fits the main assumptions of the construction given in [] and which is physically motivated. We will further consider a second example, again physically motivated, arising from the quantization of the damped harmonic oscillator, [7]. This paper is organized as follows: in the next section we introduce and discuss twodimensional pseudo-bosons analyzing some of their mathematical properties and their related coherent states. In Section III we introduce the generalized Landau levels GLL) and we discuss them in the context of Section II. Section IV is devoted to our analysis of the quantum damped harmonic oscillator, while Section V contains our conclusions. II The commutation rules In this section we will construct a two-dimensional -D) version of what originally proposed in [], to which we refer for further comments on the -D situation. Let H be a given Hilbert space with scalar product.,. and related norm.. We introduce two pairs of operators, a j and b j, j =,, acting on H and satisfying the following commutation rules [a j, b j ] =, and [a, a ] = [a, b ] = [b, b ] = 0,.) where x stands for x or x x = a j, b j ). Of course, they collapse to the CCR s for independent modes if b j = a j, j =,. It is well known that a j and b j are unbounded operators, so they cannot be defined on all of H. Following [], and writing D X) := p 0 DX p ) the common

domain of all the powers of the operator X), we consider the following: Assumption. there exists a non-zero φ 0,0 H such that a j φ 0,0 = 0, j =,, and φ 0,0 D b ) D b ). Assumption. there exists a non-zero Ψ 0,0 H such that b j Ψ 0,0 = 0, j =,, and Ψ 0,0 D a ) D a ). Under these assumptions we can introduce the following vectors in H: φ n,l = n!l! b n b l φ 0,0 and Ψ n,l = n!l! a ) n a ) l Ψ 0,0, n, l 0..) Let us now define the unbounded operators N j := b j a j and N j := N j = a j b j, j =,. It is possible to check that φ n,l belongs to the domain of N j, DN j ), and Ψ n,l DN j ), for all n, l 0 and j =,. Moreover, N φ n,l = nφ n,l, N φ n,l = lφ n,l, N Ψ n,l = nψ n,l, N Ψ n,l = lψ n,l..3) Under the above assumptions it is easy to check that Ψ n,l, φ m,k = δ n,m δ l,k Ψ 0,0, φ 0,0 for all n, m, l, k 0, which, if we chose the normalization of Ψ 0,0 and φ 0,0 in such a way that Ψ 0,0, φ 0,0 =, becomes Ψ n,l, φ m,k = δ n,m δ l,k, n, m, l, k 0..4) This means that the sets F Ψ = {Ψ n,l, n, l 0} and F φ = {φ n,l, n, l 0} are biorthogonal 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 f = Ψ n,l, f φ n,l, f H φ, h = n,l=0 φ n,l, h Ψ n,l, h H Ψ..5) What is not in general ensured is that the Hilbert spaces introduced so far all coincide, i.e. that H φ = H Ψ = H. Indeed, we can only state that H φ H and H Ψ H. However, motivated by the examples already discussed in the literature and anticipating the discussion in Section III, we make the n,l=0 Assumption 3. The above Hilbert spaces all coincide: H φ = H Ψ = H, which was introduced in []. This means, in particular, that both F φ and F Ψ are bases of H. Let us now introduce the operators S φ and S Ψ via their action respectively on F Ψ and F φ : S φ Ψ n,k = φ n,k, S Ψ φ n,k = Ψ n,k,.6) 3

for all n, k 0, which also imply that Ψ n,k = S Ψ S φ )Ψ n,k and φ n,k = S φ S Ψ )φ n,k, for all n, k 0. Hence S Ψ S φ = S φ S Ψ = S Ψ = S φ..7) 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, []. Moreover, at least formally, it is possible to write these operators in the bra-ket notation as S φ = n,k=0 φ n,k >< φ n,k, S Ψ = n,k=0 Ψ n,k >< Ψ n,k..8) 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. This aspect was exhaustively discussed in [], where the role of Riesz bases in relation with the boundedness of S φ and S Ψ has been discussed in detail. We shall come back to this aspect later. However, we shall not assume here, except when explicitly stated, what has been called Assumption 4 in [], since in most examples considered so far, and in what we are going to discuss in Section III, this assumption is not satisfied. It is interesting to remark that, as in [], even these two-dimensional pseudo-bosons give rise to interesting intertwining relations among non self-adjoint operators, see [3] and references therein. In particular it is easy to check that S Ψ N j = N j S Ψ and N j S φ = S φ N j,.9) j =,. This is related to the fact that the spectra of, say, N and N coincide and that their eigenvectors are related by the operators S φ and S Ψ, in agreement with the literature on intertwining operators, [8, 9], and on pseudo-hermitian quantum mechanics, see [0] and references therein. II. Coherent states As it is well known there exist several different, and not always equivalent, ways to define coherent states, [, ]. In this paper we will adopt the following definition, generalizing []. Recall that a set of vectors ϕ, ϕ, ϕ 3,..., is a Riesz basis of a Hilbert space H, if there exists a bounded operator V, with bounded inverse, on H, and an orthonormal basis of H, φ, φ, φ 3,..., such that ϕ j = V φ j, for all j =,, 3,... 4

Let z and z be two complex variables, z, z D some domain in C), and let us introduce the following operators: U j z j ) = e z j b j z j a j = e z j / e z j b j e z j a j, V j z j ) = e z j a j z j b j = e z j / e z j a j e z j b j,.0) j =,, and Uz, z ) := U z ) U z ), V z, z ) := V z ) V z ),.) and the following vectors: φz, z ) = Uz, z )φ 0,0, Ψz, z ) = V z, z ) Ψ 0,0..) Remarks: ) Due to the commutation rules for the operators b j and a j, we clearly have [U z ), U z )] = [V z ), V z )] = 0. ) Since the operators U and V are, for generic z and z, unbounded, definition.) makes sense only if φ 0,0 DU) and Ψ 0,0 DV ), a condition which will be assumed here. In [] it was proven that, for instance, this is so when F φ and F Ψ are Riesz bases. F φ as 3) The set D could, in principle, be a proper subset of C. It is possible to write the vectors φz, z ) and Ψz, z ) in terms of the vectors of F Ψ and φz, z ) = e z + z )/ n,l=0 z n z l n! l! φ n,l, Ψz, z ) = e z + z )/ n,l=0 z n z l n! l! Ψ n,k..3) These vectors are called coherent since they are eigenstates of the lowering operators. Indeed we can check that a j φz, z ) = z j φz, z ), b j Ψz, z ) = z j Ψz, z ),.4) for j =, and z j D. It is also a standard exercise, putting z j = r j e iθ j, to check that the following operator equalities hold: dz π dz φz, z ) >< φz, z ) = S φ, C C as well as dz π C C dz φz, z ) >< Ψz, z ) = π 5 dz π dz Ψz, z ) >< Ψz, z ) = S Ψ, C C.5) C dz C dz Ψz, z ) >< φz, z ) =,.6)

which are written in convenient bra-ket notation. It should be said that these equalities are, most of the times, only formal results. Indeed it is not difficult to construct examples in which something goes wrong and, for instance, the resolution of the identity for the pair φz, z ) and Ψz, z ) does not hold as expected. As the following theorem will show, this is a reflection of the fact that the operators S φ and S Ψ are unbounded, or, equivalently, of the fact that F φ and F Ψ are not Riesz bases. Indeed we have the following general result, which was essentially stated in [] for a concrete example of -D pseudo-bosons, and which we extend here to the general setting. Theorem Let a j, b j, F φ, F Ψ, φz, z ) and Ψz, z ) be as above. Let us assume that ) F φ, F Ψ are Riesz bases; ) F φ, F Ψ are biorthogonal. Then.6) holds true. The proof of this theorem does not differ significantly from that given in [], so that it will not be repeated here. The meaning of the theorem is the following: suppose that following the above construction the coherent states we get do not produce a resolution of the identity. Then, since F φ and F Ψ are automatically biorthogonal, they cannot be Riesz bases neither one of them)! However, this theorem does not hold in general for other types of coherent states. We will come back on this point in the next section. III Generalized Landau levels The Hamiltonian of a single electron, moving on a two-dimensional plane and subject to a uniform magnetic field along the z-direction, is given by the operator H 0 = p + Ar) ) = p x y ) + p y + x ), 3.) where we have used minimal coupling and the symmetric gauge A = y, x, 0). The spectrum of this Hamiltonian is easily obtained by first introducing the new variables P 0 = p x y/, Q 0 = p y + x/. 3.) In terms of P 0 and Q 0 the single electron hamiltonian, H 0, can be rewritten as H 0 = Q 0 + P 0 ). 3.3) 6

On a classical level, the transformation 3.) is part of a canonical map from the phase space variables x, y, p x, p y ) to Q 0, P 0, Q 0, P 0), where P 0 = p y x/, Q 0 = p x + y/, 3.4) which can be used to construct a second hamiltonian H 0 = Q 0 + P0 ). The corresponding quantized operators satisfy the commutation relations: [x, p x ] = [y, p y ] = i, [x, p y ] = [y, p x ] = [x, y] = [p x, p y ] = 0, and [Q 0, P 0 ] = [Q 0, P 0] = i, [Q 0, P 0] = [Q 0, P 0 ] = [Q 0, Q 0] = [P 0, P 0] = 0, 3.5) so that [H 0, H 0] = 0. We refer to [3] and references therein for a discussion on how the corresponding wave functions look in different representations. In [6] two of us STA and FB) have considered, in the context of supersymmetric SUSY) quantum mechanics, an extended version of H 0, an extension needed due to the fact that for the Hamiltonian of the standard Landau levels SLL) there is essentially no difference between H 0 and its SUSY partner Hamiltonian. The extension constructed in [6] is very natural and simple: introducing the vector valued function W 0 = x, y, 0) = W 0,, W 0,, 0), we may rewrite the operators in 3.) and 3.4) as P 0 = p x + W 0,, Q 0 = p y W 0,, P 0 = p y + W 0,, Q 0 = p x W 0,. 3.6) This definition was extended in [6] as follows: p = p x + W, q = p y W, p = p y + W, q = p x W, 3.7) introducing a vector superpotential W = W, W, 0). Here, since we are interested in constructing -D pseudo-bosons, it is convenient to introduce two in general) complex and different vector superpotentials this is a slight abuse of language!) W = W, W ) and V = V, V ), and we put P = p x + W, Q = p y W, P = p y + V, Q = p x V. 3.8) Our notation is the following: all operators with suffix 0 are related to the SLL. The same operators, without the 0, have to do with our generalized model, i.e. with the GLL. Notice that these operators are, in general, not self-adjoint. Hence, while for example P 0 = P 0, we 7

may have P P, depending on the choice of V. The superpotentials should also be chosen in such a way that, first of all, Q, P, Q and P satisfy the same commutation rules 3.5) as their 0-counterparts: [Q, P ] = [Q, P ] = i, [Q, P ] = [Q, P ] = [Q, Q ] = [P, P ] = 0. 3.9) These impose certain conditions on V and W : W,x = V,y, W,x = V,x, W,y = V,y, W,y = V,x, 3.0) as well as V,x + V,y = W,x + W,y =. 3.) The subscripts x, y denote differentiation with respect to that variable. Hence, as it was already clear at the beginning, the two different vector superpotentials must be related to each other. Notice that the standard choice trivially satisfies all these conditions. We now introduce the following operators: A = α Q + i P ), B = γ Q i P ), A = αq + i P ), B = γq ip ), 3.) where α γ = and α γ =. Incidentally, we recall that for the SLL the same linear combinations as in 3.) hold with α = α = γ = γ = and with the operators Q, P, Q and P replaced respectively by Q 0, P 0, Q 0 and P 0. Thus, the operators generalizing the Landau Hamiltonians in [6] are h = p x + W ) + p y W ), h = p x V ) + p y + V ), 3.3) which can be rewritten as h = B A, h = BA. 3.4) The operators in 3.) are pseudo-bosonic since they satisfy the following commutation rules: [A, B] = [A, B ] =, 3.5) while all the other commutators are trivial. It is important to observe that, since A = αq ip ), and since Q and P are not necessarily self-adjoint, in general B A. Analogously, in general B A. Similar conclusions can be deduced starting from the pairs B, A and B, A. 8

At this stage it is interesting to say few words on the physical meaning of our model. In other words: what is the physical meaning of going from the SLL to these GLL? The answer is the following: suppose we interpret W and V in 3.3) as two different but related) vector potentials describing two possibly different magnetic fields. These potentials are A = W, W, 0) for h and A = V, V, 0) for h the reason for this notation will be clear in a moment). Now, computing the associated magnetic fields from these vectors we get B = A = ˆk x W + y W ) = ˆk, B = A = ˆk x V + y V ) = ˆk, because of the equalities in 3.). Hence, for any possible choice of superpotentials, h and h respectively describe an electron in an up and in a down uniform magnetic field, as the original hamiltonians H 0 and H 0. Incidentally this suggests that we should further analyze this model in the light of the modular structure, recently considered in [4] in the context of SLL. The following are some possible choices of W and V : Choice SLL). Let us take V x, y) = W x, y) = x, V x, y) = W x, y) = y. If we further take α = γ = α = γ = we recover exactly the usual situation, [6]. Moreover, we go back to bosonic rather than pseudo-bosonic commutation relations. Choice Perturbations of the SLL). First we consider a symmetric perturbation. For that we take V x, y) = x + v y), V x, y) = y + v x), where v and v are arbitrary but sufficiently regular) functions. Hence we get, apart from inessential additive constants, W x, y) = x v y), W x, y) = y v x). In order not to trivialize the situation, it is also necessary to take v y) and v x) complex at least one of them): this is the way to get pseudo-bosons rather than simple bosons. A non symmetric version of this perturbation can be constructed by just taking V x, y) = a x + v y), V x, y) = a y + v x), with a + a =. Choice 3 A general solution). We take V x, y) = x + v y) + V x,y) dx, where y V x, y) is any function for which this definition makes sense. In particular, for instance, if we take V x, y) = e xy then V x, y) = x + v y) + x y ) e xy and, consequently, y W x, y) = v y) x y ) e xy and W y x, y) = y e xy. If we rather take V x, y) = x n y k, n, k =,, 3,..., then V x, y) = x+v y) k n+ xn+ y k, and so on. 9

III. A perturbation of the SLL We will now focus our attention on Choice above, with an explicit choice of v y) and v x), and apply the construction given in Section II. Let W x, y) = x ik y, W x, y) = y ik x, 3.6) with k and k real and not both zero not to go back to SLL). In this case the operators in 3.) assume the following differential expressions: A = α x i y + x + k ) iy k ) ), B = γ x i y + x k ) + iy + k ) ), A = α i x + y ix + k ) + y k ) ), B = γ i x y + ix k ) + y + k ) ). 3.7) In order to check Assumptions and of the previous section, we first look for vectors φ 0,0 x, y) and Ψ 0,0 x, y) satisfying Aφ 0,0 x, y) = 0 and B Ψ 0,0 x, y) = 0. We get { } φ 0,0 x, y) = N φ exp x + k 4 ) y k 4 ) { } Ψ 0,0 x, y) = N Ψ exp x k 4 ) y + k 4 ), 3.8) where N φ and N Ψ are normalization constants which are chosen in such a way that φ 0,0, Ψ 0,0 =. Of course, in order for this result to make sense, the two functions must belong to the Hilbert space H we are considering here, i.e. L R ). This imposes some constraints on k and k : < k j <, j =,. It is possible to check that the same functions also satisfy A φ 0,0 x, y) = 0 and B Ψ 0,0 x, y) = 0. It is now evident that Assumptions and are satisfied. Indeed the action of, say, B n on φ 0,0 x, y) simply produces some polynomial see 3.) below) of the n-th degree times a gaussian: this resulting function belongs clearly to L R ) for all n. This fact allows us to define the following functions φ n,l x, y) = B n B l n! l! φ 0,0 x, y), and Ψ n,l x, y) = A ) n A ) l n! l! Ψ 0,0 x, y), 3.9) where n, l = 0,,, 3,.... As we have seen in the previous section, the sets F Ψ = {Ψ n,l x, y), n, l 0} and F φ = {φ n,l x, y), n, l 0} are biorthogonal. In fact, with our previous choice of the normalization constants, we have Ψ n,l, φ m,k = δ n,m δ l,k, n, m, l, k 0. 3.0) 0

Of course these vectors diagonalize the operators h = N and h = N, as well as their adjoints h = N and h = N, where N = BA, N = B A, N = N and N = N. We find: h φ n,l = n ) φ n,l, h φ n,l = l ) φ n,l, and h Ψ n,l = n ) Ψ n,l, h Ψ n,l = l ) Ψ n,l. The next step consists in proving that the sets F φ and F Ψ are complete in H. This is a consequence of the fact that a.) the set F h := {h n,m x, y) := x n y m φ 0,0 x, y), n, m 0} is complete in L R ); b.) each function of F h, can be written as a finite linear combination of some φ i,j x, y). Then it is clear that, if by assumption f H is such that f, φ i,j = 0 for all i and j, then f, h n,m = 0 for all n and m, so that f = 0. Of course the same argument allows us to prove that F Ψ is complete in H. This result implies that also Assumption 3 of Section II is satisfied. Now we could introduce the intertwining operators S φ and S Ψ and check, among other properties, if they are bounded or not. This is related to the fact that, as we will first show, the sets F φ and F Ψ are not Riesz bases, except when k = k = 0 see 3.8)). To check this claim, we introduce the orthonormal basis of L R ) arising from the SLL, [6], { F φ 0) := φ 0) n,l x, y) := B n 0 B l 0 n! l! φ 0) 0,0x, y), n, m 0 where φ 0) 0,0x, y) = π e x +y )/4 is the vacuum of A 0 = Q 0 + ip 0 ) and A 0 = Q 0 + ip 0). Recall that, for SLL, B 0 = A 0 and B 0 = A 0. To prove now that F φ is not a Riesz basis, we will show that an operator T φ exists mapping into F φ, that T φ is invertible, but T φ and/or Tφ is simple. Indeed it is easy to first check that F 0) φ and φ 0) n,0x, y) = n n! x + iy)n φ 0) }, are not bounded. Finding this operator 0,0x, y), φ 0) il 0,l x, y) = l l! x iy)l φ 0) 0,0x, y) 3.) φ n,0 x, y) = γ n n! x + iy) n φ 0,0 x, y), φ 0,l x, y) = iγ)l l! x iy) l φ 0,0 x, y), 3.) for all n, l 0. Similar formulae are deduced for Ψ n,0 x, y) and Ψ 0,l x, y). From a comparison between 3.) and 3.) it is clear that T φ can exist only if γ = γ =. Assuming this to

be so, we have for all n, l 0. equality: φ n,0 x, y) φ 0) n,0x, y) = φ 0,lx, y) φ 0) 0,l x, y) = φ 0,0x, y) φ 0) 0,0x, y), 3.3) This suggest that we define T φ as the ratio in the right-hand side of this T φ = φ 0,0x, y) φ 0) 0,0x, y) = πnφ e x k + y k. 3.4) Of course we have still to check that with this definition φ n,l x, y) = T φ φ 0) n,l x, y) holds also if both n and l are not zero. This can be proven observing that, for all n 0, the following intertwining relation holds: Therefore, since B n T φ = T φ A 0 ) n. 3.5) φ n,l x, y) = T φ φ 0) n,l x, y) B n φ 0,l = T φ A 0 ) n φ 0) 0,l B n T φ φ 0) 0,l = T φ A 0 ) n φ 0) 0,l, our claim immediately follows. Formula 3.5) can be proved by induction on n. The inverse of T φ is Tφ = πnφ e x k y k. It is clear that both T φ and/or Tφ are unbounded on L R ) for all possible choices of k and k in, ), except when k = k = 0, i.e., in the case of the SLL. Hence, for well known general reasons, [5, 6], F φ cannot be a Riesz basis. Essentially the same arguments also show that F Ψ is not a Riesz basis, either. Indeed, an operator T Ψ mapping F φ 0) into F Ψ can be found and it is T Ψ = Ψ 0,0x, y) φ 0) 0,0x, y) = πnψ e x k y k. 3.6) This operator satisfies Ψ n,l x, y) = T Ψ φ 0) n,l x, y) for all possible choices of n and l greater or equal to zero. Therefore, since φ n,l x, y) = T φ φ 0) n,l x, y) = T φt Ψ )Ψ n,lx, y), the operators S φ and S Ψ in.6) can be easily identified and look like S φ = T φ T Ψ = N φ N Ψ e x k +y k, S Ψ = S φ = T Ψ T φ = N Ψ N φ e x k y k. 3.7) Notice that for any choice of k and k in, ), other than when k, k ) = 0, 0), at least one of these operators is unbounded. We will now construct a set of bicoherent states for our GLL. However, rather than using the definitions in.), it is convenient to look for solutions in the x, y) space of the eigenvalue

equations A φx, y; z, z ) = z φx, y; z, z ) A φx, y; z, z ) = z φx, y; z, z ) B Ψx, y; z, z ) = z Ψx, y; z, z ) B Ψx, y; z, z ) = z Ψx, y; z, z ) 3.8) where, as suggested by our previous results, we take α = α = γ = γ = in 3.7). The square integrable solutions of the differential equations in 3.8) are { φx, y; z, z ) = N A z, z ) e [+k )x k )y ]/4 e [z +iz)x+z+iz )y] Ψx, y; z, z ) = N B z, z ) e [ k )x +k )y ]/4 e [z +iz)x+z+iz )y], 3.9) where z and z are complex parameters. The normalization is fixed by requiring that φx, y; z, z ), Ψx, y; z, z ) L R ) = φz, z ), Ψz, z ) H, where in the rhs the coherent states introduced in.), and living in the Hilbert space H, appear. Notice that φz, z ), Ψz, z ) H = for all z and z in D. Then we find, with a suitable choice of phases, N A z, z )N B z, z ) = e z iz. π Notice that these states reduce to the standard two dimensional gaussian π e x +y )/4 when z = z = k = k = 0, i.e., for the SLL and for eigenvalues of the lowering operators both equal to zero. It is now a straightforward computation to check the resolution of the identity for these states where is the identity in L R ). dzdz φx, y; z, z ) >< π Ψx, y; z, z ) = 3.30) C This result is by no means in disagreement with the theorem stated in Section II. The first reason is that it is not clear that the functions φx, y; z, z ) and Ψx, y; z, z ) coincide with φz, z ) and Ψz, z ), for which the theorem was stated. Secondly, and more important, that theorem gives only a sufficient condition. Hence, if we would be able to prove that φx, y; z, z ) = φz, z ) and Ψx, y; z, z ) = Ψz, z ), this computation will provide a nice counterexample showing that the conditions of the theorem are, in fact, only sufficient and not necessary. This is work in progress. 3

IV Damped harmonic oscillator An interesting example of two-dimensional pseudo-bosons is provided by the damped harmonic oscillator DHO). In [7] the authors have discussed a possible approach to the quantization of the DHO. This is a non conservative system, so that a Hamiltonian approach requires a certain amount of care. The approach which was proposed already in 977, [7], is to consider the DHO as a part of a larger system, involving also a second oscillator which is forced and which takes the energy lost by the DHO, so that this larger system is conservative. The original equation of motion, mẍ + γẋ + kx = 0, is therefore complemented by a second virtual equation, mÿ γẏ + ky = 0, and the classical lagrangian for the system looks like L = mẋẏ+ γ xẏ ẋy) kxy, which ) ) corresponds to a classical Hamiltonian H = p x ẋ + p y ẏ L = px + γ y py γ x m + kxy, where p x = L and p ẋ y = L are the conjugate momenta. The introduction of pseudo-bosons ẏ is based on two successive changes of variables and on a canonical quantization. First of all we introduce the new variables x and x via x = x + x ), y = m ẋ ẋ )+ γ x ẋ x ẋ ) k x x ) and H = m The second change of variable is the following: where we have introduced Ω = p + = ω + p mω + i mωω + mωω p γ x ) + m x, x x ). Then L = ) p + γ x + k x x ). p = ω p mω i x, 4.) mω x + = ω + x + i mωω + p, mω x = ω x i mωω p, ) k γ and the two following complex quantities ω m 4m ± = Ω ± i γ γ. In the rest of the section we will assume that k, so that Ω is real. Up to now, m 4m we are still at a classical level, so that ω + = ω, p + = p, x + = x, and consequently, see below, H + = H and H = H. Hence H is a real Hamiltonian. Indeed, with these definitions, the Hamiltonian looks like the hamiltonian of a two-dimensional harmonic oscillator at least formally. H = p + + ω+x+) ) + p + ω x =: H+ + H At this stage we quantize canonically the system, [7]: we require that the following commutators are satisfied: [x +, p + ] = [x, p ] = i, 4.) 4

all the other commutators being trivial. We also have to require that p + = p and that x + = x, which are the quantum version of the compatibility conditions above. The pseudo-bosons now appear: a + = ) ω + x + + i p + ω +, a = ) ω x + i p ω, b + = ) ω + x + i p + ω +, b = ) ω x i p ω, and indeed we have [a +, b + ] = [a, b ] =, all the other commutators being zero. Notice also that b + = a and b = a +. Moreover H can be written in term of the operators N ± = b ± a ± as H = ω + N + + ω N + ω ++ω. So the hamiltonian of the quantum DHO is simply written in terms of pseudo-bosonic operators. 4.3) IV. About Assumptions -3 This system provides a non trivial example of pseudo-bosonic operators which do not satisfy any of the Assumptions -3 of Section II. To show this, we first observe that a possible representation of the operators in 4.) is the following x + = Γ δ δ Γ Γ py + δ x ), x = Γ p Γ δ δ Γ y + δ x), p + = Γ p x + δ y, p = Γ p x + δ y, for all choices of Γ and δ such that Γ δ δ Γ. Here x, y, p x and p y are pairwise conjugate self-adjoint operators: [x, p x ] = [y, p y ] = i. 4.4) Notice that these operators also satisfy the compatibility conditions p + = p and x + = x. Hence it is natural to represent x and y as the standard multiplication operators and p x and p y as i x = i x and i y = i y. Then we get a + = ω + a = ω b + = ω + b = ω { ) )} β x + i δ Γ ω + y + ω + x i α y, { ) )} β x + i δ Γ ω y + ω x i α y, { ) )} β x i δ Γ ω + y ω + x + i α y, { ) )} β x i δ Γ ω y ω x + i α y, 4.5) 5

where, to simplify the notation, we have introduced α = Γ and β = Γ δ δ Γ δ. Γ δ δ Γ Remark: a different representation of x ± and p ± could be deduced using the results of Section III. However, while the pseudo-bosonic commutation rules would be easily recovered, the compatibility conditions x + = x and p + = p would be lost. Hence this choice is not compatible with our requirements. Assumption of Section II requires the existence of a square-integrable function φ 0,0 x, y) such that, first of all, a + φ 0,0 x, y) = a φ 0,0 x, y) = 0. Analogously, Assumption requires the existence of a possibly different) square-integrable function Ψ 0,0 x, y) such that, first of all, b +Ψ 0,0 x, y) = b Ψ 0,0 x, y) = 0. However, since b + = a and b = a +, these two functions, if they exist, satisfy the same differential equations. Hence, apart from a normalization constant, we can chose them to be coincident. It is possible to check that a solution of a + φ 0,0 x, y) = a φ 0,0 x, y) = 0 is the following: { φ 0,0 x, y) = N 0 exp β ω + Γ x + δ } y. 4.6) α ω + Notice that, in order for this function to be a solution of both a + φ 0,0 x, y) = 0 and a φ 0,0 x, y) = 0 it is necessary and sufficient to have the following identity satisfied: ω + ω = δ Γ. This is not δ Γ a big requirement, clearly. What is crucial, on the other hand, is that the function φ 0,0 x, ) y), and Ψ 0,0 x, y) should consequently be square integrable. This is possible only if R β ω+ > 0 Γ ) δ and if, at the same time, R α ω + < 0. Now, it is not hard to check that these two conditions are incompatible: if one is verified, the other is not. Therefore the conclusion is that, following the procedure we have considered so far, Assumptions and are violated and, of course, Assumption 3 cannot even be considered since it is meaningless. Of course this does not mean that for the quantum DHO the construction proposed in Section II cannot be considered. It only means that with the choices we have considered here, this is not possible. It could be possible, however, to look for some different representation of the operators, satisfying the compatibility condition, and see if it is possible to satisfy Assumptions, and 3. This is work in progress. V Conclusions In this paper we have constructed a physically motivated two-dimensional family of pseudobosons arising from a generalized version of the Landau levels. This generalization has been 6

shown to be essentially a gauge transformation. Coherent states have been constructed and the resolution of the identity has been proved. We have also considered a quantum damped harmonic oscillator: this provides a nice example of a pseudo-bosonic system for which all the assumptions of Section II are violated. In conclusion, many examples exist, see Section III and references [,, 5, 8] among the others, in which Assumptions -3, and sometimes Assumption 4, are satisfied. But other examples exist as well for which, even if pseudo-bosonic commutation rules are recovered, none of the Assumptions hold true. This suggests to take care explicitly of these Assumptions when dealing with pseudo-bosons. Acknowledgements The authors would like to acknowledge financial support from the Università di Palermo through Bando CORI, cap. B.U. 9.3.000.000.000. One of us STA) would like to acknowledge a grant from the Natural Sciences and Engineering Research Council NSERC) of Canada. References [] F. Bagarello, Pseudo-bosons, Riesz bases and coherent states, J. Math. Phys., 50, DOI:0.063/.3300804, 0353 00) 0pg) [] F. Bagarello Construction of pseudo-bosons systems, J. Math. Phys., 5, doi:0.063/.3300804, 0353 00) 0pg) [3] F. Bagarello Mathematical aspects of intertwining operators: the role of Riesz bases, J. Phys. A, doi:0.088/75-83/43/7/7503, 43, 7503 00) pp) [4] F. Bagarello, Examples of Pseudo-bosons in quantum mechanics, Phys. Lett. A, in press [5] D.A. Trifonov, Pseudo-boson coherent and Fock states, quant-ph/090.3744 [6] Ali S.T., Bagarello F., Supersymmetric associated vector coherent states and generalized Landau levels arising from two-dimensional supersymmetry, J. Math Phys., 49, DOI: 0.063/.8987, 008) [7] R. Banerjee, P. Mukherjee A canonical approach to the quantization of the damped harmonic oscillator, J. Phys. A, 35, 559 00) 7

[8] Kuru S., Tegmen A., Vercin A., Intertwined isospectral potentials in an arbitrary dimension, J. Math. Phys, 4, No. 8, 3344-3360, 00); Kuru S., Demircioglu B., Onder M., Vercin A., Two families of superintegrable and isospectral potentials in two dimensions, J. Math. Phys, 43, No. 5, 33-50, 00); Samani K. A., Zarei M., Intertwined hamiltonians in two-dimensional curved spaces, Ann. of Phys., 36, 466-48, 005). [9] F. Bagarello Extended SUSY quantum mechanics, intertwining operators and coherent states, Phys. Lett. A, DOI: 0.06/ j.physleta. 008.08.047 008), F. Bagarello Vector coherent states and intertwining operators, J. Phys. A., doi:0.088/75-83/4/7/07530, 009), F. Bagarello, Intertwining operators between different Hilbert spaces: connection with frames, J. Math. Phys., DOI: 0.063/.3094758, 50, 043509 009) 3pp) [0] A. Mostafazadeh, Pseudo-hermitian quantum mechanics, quant-ph/080.5643, C. Bender, Making Sense of Non-Hermitian Hamiltonians, Rep. Progr. Phys., 70, 947-08 007) [] S.T. Ali, J-P. Antoine and J-P. Gazeau, Coherent States, Wavelets and Their Generalizations, Springer-Verlag, New York, 000. [] J-P. Gazeau, Coherent states in quantum physics, Wiley-VCH, Berlin 009 [3] J.P. Antoine, F. Bagarello, Localization properties and wavelet-like orthonormal bases for the lowest Landau level, in Advances in Gabor Analysis, H.G. Feichtinger, T. Strohmer, Eds., Birkhäuser, Boston, 003. [4] S.T. Ali, F. Bagarello, G. Honnouvo, Modular Structures on Trace Class Operators and Applications to Landau Levels, J. Phys. A, doi:0.088/75-83/43/0/050, 43, 050 00) 7pp) [5] Young R., An introduction to nonharmonic Fourier series, Academic Pree, New York, 980) [6] Christensen O., An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 003) [7] H. Feshbach, Y. Tikochinsky, N.Y. Acad. Sci., 38, 44 977) [8] F. Bagarello, Examples of Pseudo-bosons in quantum mechanics, Phys. Lett. A, 374, 383-387 00) 8