Non-isospectral Hamiltonians, intertwining operators and hidden hermiticity
aa r X i v : . [ m a t h - ph ] O c t Non-isospectral Hamiltonians, intertwining operatorsand hidden hermiticity
F. Bagarello
Dipartimento di Metodi e Modelli Matematici, Facolt`a di Ingegneria,Universit`a di Palermo, I-90128 Palermo, Italye-mail: [email protected]
Abstract
We have recently proposed a strategy to produce, starting from a given hamiltonian h and a certain operator x for which [ h , xx † ] = 0 and x † x is invertible, a second hamiltonian h with the same eigenvalues as h and whose eigenvectors are related to those of h by x † . Here we extend this procedure to build up a second hamiltonian, whose eigenvaluesare different from those of h , and whose eigenvectors are still related as before. This newprocedure is also extended to crypto-hermitian hamiltonians. Introduction
The problem of finding quantum system for which the Schr¨odinger equation can be solvedexactly is not an easy task: obtaining the eigenvalues and the eigenvectors of a given hamiltonianis often possible only at a perturbative level. Hence, finding new potentials for which somenon-perturbative solution can be found is a rather useful goal in theoretical physics, whichhas produced many results in the so-called super-symmetric quantum mechanics (SUSY qm),[1], and in the theory of intertwining operators (IO), see [2] and references therein. Recently,[3, 4, 5], we have proposed a simple technique which produces, starting from a given hamiltonian h with known eigensystem, a second hamiltonian h with the same eigenvalues and eigenvectorswhich are easily deduced from those of h . In other words, this means that we produce, froma solvable potential , a second quantum potential which is also solvable. This is what, in theliterature, is called a Darboux transform . Our technique relies, in particular, on the possibilityof inverting a certain operator N , see below. In this paper we show that, if N is not invertible,the same general strategy could be used to construct, out of h , a second hamiltonian, which weagain call h , whose eigenstates and eigenvalues are different from those of h but which howevercan be easily computed from these ones. Moreover, within our new results, the requirement of h to be self-adjoint is not really necessary and can be replaced by its crypto-hermiticity, in thesense of Znojil, [6]: h = Θ − h † Θ, for a certain operator Θ. In this case, we will show that theprocedure discussed in Section II can be easily modified and several interesting results can stillbe obtained. In particular, we will see how to construct several sets of complete vectors in theHilbert space H , as well as several intertwining operators. It is maybe useful to stress here thatcrypto-hermitian quantum mechanics and its many relatives , see [7] and references therein, isnowadays a rather hot topic, both for its theoretical aspects and for some recent experimentswhich seem to fit well in this new scheme, [8].This paper is organized as follows: in the next section we discuss the general strategy andsome of its consequences. In Section III we discuss two examples in which the eigenvalues of h have multiplicity one. In Section IV we give examples with multiplicity larger than one. InSection V we extend our construction to crypto-hermitian hamiltonians. Section VI containsour conclusions. 2 I The strategy
In a recent paper the possibility of constructing, from a self-adjoint hamiltonian h , a secondhamiltonian with the same eigenvalues and related eigenvectors have been discussed in detail,[3, 4, 5]. Let h be a self-adjoint hamiltonian on the Hilbert space H , h = h † , whose (notnecessarily normalized) eigenvectors, ϕ (1) n , satisfy the following equation: h ϕ (1) n = ǫ n ϕ (1) n , n ∈ N := N ∪ { } . Let H be a second Hilbert space, in general different from H , and let usconsider an operator x : H → H , whose adjoint x † maps H in H . Let us further define N := x x † and N := x † x . These are surely well defined if x is a bounded operator. Onthe other hand, if x is unbounded, N is well defined, if, taken f in the domain of x , D ( x ), xf ∈ D ( x † ). Analogously, N is well defined if, taken g in D ( x † ), x † g belongs to D ( x ). It isclear that N j maps H j into itself, for j = 1 ,
2. Suppose now that x is such that N is invertiblein H and that [ N , h ] = 0. Of course, this commutator should be considered in a weak form if h or N is unbounded: < N f, h g > = < h f, N g > , for f, g ∈ D ( N ) ∩ D ( h ). Here <, > is the scalar product in H . Defining now˜ h := N − (cid:0) x † h x (cid:1) , ϕ (2) n = x † ϕ (1) n , (2.1)in [3, 4, 5] it is shown that ˜ h is self-adjoint, ˜ h = ˜ h † , that it satisfies the following modifiedversion of intertwining relation x † (cid:16) x ˜ h − h x (cid:17) = 0 and that, if ϕ (2) n = 0, then ˜ h ϕ (2) n = ǫ n ϕ (2) n .Also, [ N , ˜ h ] = 0, again in a weak form, in general. Furthermore, if ǫ n is non degenerate, ϕ (1) n and ϕ (2) n are respectively eigenstates of N and N with the same eigenvalue.In [3, 4, 5] we have proposed several examples of this construction, some arising from thetheory of the (g)-frames and some from quons, [9]. In particular, we have shown that it isconvenient, if possible, to avoid any explicit representation of the operators involved in thedefinition of h and x and work, as much as possible, at a purely algebraic level, playingwith the commutation relations. The main difficulty in the cited papers is the computation of N − , which however is rarely needed (in the examples considered) since it usually disappearsafter some re-ordering of the operators of ˜ h . But when this is not so, then some Green’sfunction should be computed and this may not be a simple task, from a practical point ofview. Therefore, the computation of N − makes our strategy difficult to be applied. More thanthis, it could also happen that N has no inverse at all! Hence a natural question to answeris the following: if N − does not exist, or if we are not able to compute it, are we still ableto deduce a second quantum system whose eigenvalues and eigenvectors can be easily found?Luckily enough, the answer is affirmative, and this can be done with a simple extension of theidea sketched above. 3et h be, as before, a self-adjoint hamiltonian on the Hilbert space H , h = h † , withnormalized eigenvectors ϕ (1) n,k : h ϕ (1) n,k = ǫ (1) n ϕ (1) n,k , n ∈ N and k = 1 , , . . . m n , m n being thedegeneracy of ǫ n . To simplify the notation, we will call J the set of these quantum numbers.Let H be a second Hilbert space, in general different from H , and let x be an operator from H to H . In this paper we will mainly consider the case of x bounded, but quite often we will alsocomment on what happens for unbounded x . Let us further define N := x x † , N := x † x . Asalready discussed, these operators are well defined if x is bounded while some extra requirementhas to be assumed for unbounded x . We assume here first that, for such x , [ N , h ] = 0 (ina weak sense, if needed). Nothing will be assumed on the existence of N − . Because of thecommutativity between h and N F = n ϕ (1) n,k , ( n, k ) ∈ J o can be taken to be a family ofeigenstates of N as well, N ϕ (1) n,k = ν n,k ϕ (1) n,k , ∀ ( n, k ) ∈ J . We will assume here that, using thelanguage of physicists, h and N are a complete set of commuting observables . In other words,the set F is a basis of H , which is clearly orthonormal (o.n.): D ϕ (1) n,k , ϕ (1) m,l E = δ n,m δ k,l . This isnot a big requirement since, if it is not true, we could always replace the original Hilbert space H with a new one, ˜ H , constructed taking the closure of the linear span of F . The closurerelation of F in bra-ket language reads as follows: X ( k,n ) ∈ J (cid:12)(cid:12)(cid:12) ϕ (1) n,k ih ϕ (1) n,k (cid:12)(cid:12)(cid:12) = 11 , (2.2)where 11 is the identity operator in H . Due to the definition of N it is clear that its eigenvalues ν n,k cannot be negative. Indeed we find that ν n,k = k x † ϕ (1) n,k k , k . k being the norm in H , whichis always positive and is zero if and only if ϕ (1) n,k ∈ ker( x † ). Therefore, if ker( x † ) = { } , then ν n,k > n, k ) ∈ J and, as a consequence, N admits inverse.Let us now define h := x † h x, ϕ (2) n,k := x † ϕ (1) n,k , ǫ (2) n,k := ǫ (1) n ν n,k . (2.3)Notice that, in principle, ϕ (2) n could be zero if ker( x † ) = { } . To begin with, the followingproperties can be easily deduced: h = h † , [ h , N ] = 0 , N x = x N , ( h N ) x = x h . (2.4)With our definitions, therefore, h is also self-adjoint and commutes with N (weakly, if needed).From this point of view, h and N behave exactly as h and N . Moreover, x intertwinesbetween N and N , as well as between h N and h , and this will have consequences on the4elated eigensystems. In particular we have that, if ϕ (1) n,k / ∈ ker( x † ), then the non zero vector ϕ (2) n,k satisfies the following eigenvalue equations h ϕ (2) n,k = ǫ (2) n,k ϕ (2) n,k , N ϕ (2) n,k = ν n,k ϕ (2) n,k , (2.5)whose proofs are straightforward. Formula (2.5) shows a first difference between h and h :while the first has degenerate eigenvalues, in general, the eigenvalues of h are not degenerate.In view of what has been discussed before we can also say that ν n,k = 0 if and only if ϕ (2) n,k = 0.Hence, if ν n,k >
0, we can deduce that ϕ (2) n,k / ∈ ker( x ) and, more than this, that ϕ (1) n,k = 1 ν n,k x ϕ (2) n,k , (2.6)which, in a sense, reverse the second equation in (2.3).Let now J ′ = { ( n, k ) ∈ J : ν n,k > } . A consequence of the orthonormality of the set F is that also the functions of F = n ϕ (2) n,k , ( n, k ) ∈ J ′ o are orthogonal but not normalized, ingeneral. Indeed we have, taking ( n, k ) , ( m, l ) ∈ J ′ , D ϕ (2) n,k , ϕ (2) m,l E = D x † ϕ (1) n,k , x † ϕ (1) m,l E = D N ϕ (1) n,k , ϕ (1) m,l E = ν n,k δ n,m δ k,l . Here, obviously, <, > is the scalar product in H . Let us now prove the following result: Proposition 1 ker( x ) = 0 if and only if F is complete in H . Proof –
We divide the proof of the statement in two parts: J = J ′ and J ′ ⊂ J . First case: J = J ′ . As we have already discussed, since J = J ′ , ν n,k > n, k ) ∈ J . Let us prove firstthat, if ker( x ) = 0 then F is complete in H . For that we consider a vector f ∈ D ( x ) ⊆ H orthogonal to all the ϕ (2) n,k ’s: D f, ϕ (2) n,k E = 0, ∀ ( n, k ) ∈ J . Here the domain of x , D ( x ), can betaken coincident with H itself if x is bounded. Otherwise D ( x ) is dense in H ; we want todeduce that f = 0. First we notice that xf = 0, since F is complete in H by assumption.But ker( x ) = { } . Hence f = 0. This ends the proof for x bounded, while, if x is unbounded,we simply use the density of D ( x ) in H .Let us now suppose that F is complete. We will show that ker( x ) = { } . Let f ∈ ker( x ).Hence xf = 0. Since xf ∈ H and since F is an o.n. basis of H we have xf = X ( n,k ) ∈ J D xf, ϕ (1) n,k E ϕ (1) n,k = X ( n,k ) ∈ J D f, ϕ (2) n,k E ϕ (1) n,k , ϕ (2) n,k and the fact that J = J ′ . Then, taking the scalarproduct of the above expansion with ϕ (1) m,l and recalling that xf = 0, we deduce that, ∀ ( n, k ) ∈ J ,0 = D f, ϕ (2) m,l E , which implies that f = 0 since F is complete by assumption. Hence ker( x ) = { } . Second case: J ′ ⊂ J . To be concrete, we will assume here that ν , = 0, while all the others ν n,k are strictlypositive. Let us prove first that, also in this case, if ker( x ) = { } , then F is complete. Let usassume that D f, ϕ (2) n,k E = 0 for all ( n, k ) ∈ J ′ . This implies that xf = αϕ (1)0 , , for some complex α . Hence, x † x f = α x † ϕ (1)0 , = 0, since ν , = 0, so that (cid:10) f, x † x f (cid:11) = k x f k = 0. This impliesthat f ∈ ker( x ), which only contains the zero vector by assumption. Hence f = 0 and F iscomplete, as a consequence.Let us now assume that F is complete. Then, if f ∈ ker( x ), xf = 0. Repeating thesame steps as before we have 0 = xf = P ( n,k ) ∈ J ′ D f, ϕ (2) n,k E ϕ (1) n,k , which obviously implies that D f, ϕ (2) m,l E = 0 for all ϕ (2) m,l ∈ F . But F is complete. Hence f = 0 and ker( x ) = { } . (cid:3) Remarks:– (1) This Proposition is a somehow refined version of a similar result containedin [10]. In particular we are here considering the possibility of x to be unbounded, and thepossibility that not all the ν n,k are strictly positive.(2) There is an evident asymmetry between ker( x ) and ker( x † ) in this Proposition. Thereason is clear: we are assuming that F is an o.n. basis of H and we have shown that, ifker( x ) = { } , F is also a basis of H . The role of ker( x ) and ker( x † ) would be exchanged if westart with the assumption that F is a basis and we ask for conditions which makes of the set F of vectors (2.6) a basis of H . III Examples with multiplicity 1
III.1 standard bosons
Let a be the usual annihilation operator satisfying [ a, a † ] = 11, and let ϕ be the vacuum of a : aϕ = 0. Then the set F := n ϕ n := √ n ! a † n ϕ , n ≥ o is an o.n. basis of H = H = H .Let us further define h = a † a =: ˆ n , the number operator. Hence ϕ (1) n ≡ ϕ n and ǫ (1) n = n : h ϕ (1) n = nϕ (1) n , n = 0 , , , . . . . If we now take x := a k , for a fixed natural k , it is clear that6er( x ) = { ϕ , ϕ , . . . , ϕ k − } , so that the set F will not be complete in H . This can be seenexplicitly since the vectors of F are the following: ϕ (2)0 = x † ϕ (1)0 = √ k ! ϕ (1) k , ϕ (2)1 = x † ϕ (1)1 = p ( k + 1)! ϕ (1) k +1 , and so on. It is clear that, for instance, the non zero vector ϕ (1)0 is orthogonalto all the ϕ (2) n ’s, so that F cannot be complete. On the other hand, we can also check thatker( x † ) = { } . The operator N = xx † = a k a † k commutes with h for all k , as can be checkedusing induction on k . Moreover, it is interesting to notice that N = a † k a k admits no inverse.Hence, the procedure proposed in [3] does not apply here.The hamiltonian h = x † h x = a † k +1 a k +1 can be written in terms of the number operatorˆ n as follows: h = (ˆ n − k
11) (ˆ n − ( k − · · · (ˆ n − n −
11) ˆ n, (3.1)whose proof can be again given by induction. The vectors ϕ (2) n = x † ϕ (1) n = q ( k + n )! n ! ϕ n + k areeigenstates of h with eigenvalues ǫ (2) n := ( n + k )!( n − . Not surprisingly, h , ϕ (2) n and ǫ n all depend on k , which is a consequence of the fact that x itself depends on k . We also find N ϕ (1) n = ν n ϕ (1) n , ν n = ( n + k )! n ! , so that, as expected, ν n ǫ (1) n = ǫ (2) n . Moreover N = x † x coincides with h with k replaced by k −
1. Hence it is clear that [ h , N ] = 0. It is a simple exercise to check that all the propertieslisted in Section II are satisfied. III.2 generalizing this example
Following [5] we consider two operators, B and B † , which satisfy the modified commutationrelation [ B, B † ] q := B B † − qB † B = 11, q ∈ [0 , ϕ (1)0 be the vacuum of B : Bϕ (1)0 = 0. Letfurthermore h = B † B . Then, putting ϕ (1) n = 1 β · · · β n − B † n ϕ (1)0 = 1 β n − B † ϕ (1) n − , n ≥ , (3.2)we have h ϕ (1) n = ǫ (1) n ϕ (1) n , with ǫ (1)0 = 0, ǫ (1)1 = 1 and ǫ (1) n = 1 + q + · · · + q n − for n ≥
1. Also,the normalization is found to be β n = 1 + q + · · · + q n , for all n ≥
0. Hence ǫ (1) n = β n − for all n ≥
1. The set of the ϕ (1) n ’s spans the Hilbert space H = H = H and they are mutually o.n.: < ϕ (1) n , ϕ (1) k > = δ n,k .We now take, as for the ordinary bosons discussed before, x = B k . Again, its kernel isdifferent from the zero vector. Hence F is not complete. The operator N = B k B † k commuteswith h for all fixed k . This can be seen, for instance, using the following recurrence relation:7 ( k +1)1 = N ( k )1 (cid:16) q k N (1)1 + ǫ (1) k (cid:17) , where we have introduced the suffix k to make explicit thedependence on k here. The eigenvalues of N = N ( k )1 obey the following recurrence rule: ν (1) n = 1 + qǫ (1) n , ν ( k +1) n = ν ( k ) n ( q k +1 ǫ (1) n + ǫ (1) n +1 ). Notice that, in the limit q → − , we find ǫ (1) n → n , ν (1) n → n .With the usual definitions we find that h = ( B † ) k +1 B k +1 and that ϕ (2) n = x † ϕ (1) n coincides,but for a normalization, with ϕ (1) n + k (which again shows that F is not complete). It is againa matter of simple but boring computations to check that all the properties of the previoussection are satisfied. IV Examples from Landau levels
Let us consider an electron in a uniform magnetic field oriented in the positive z -direction, withvector potential ~A ↑ = B ( − y, x, H ↑ = m (cid:16) ~p − ec ~A ↑ (cid:17) , can be written as H ↑ = H + H ↑ = ~ ω ( N + + N − + 11) + ~ ω ( N − − N + ) = ~ ω (2 N − + 11) . (4.1)Here we have introduced ω = eB mc , a x = p mω ~ x + i √ mω ~ p x , a y = p mω ~ y + i √ mω ~ p y , A ± = √ ( a x ∓ ia y ) and N ± = A †± A ± . These operators satisfy the canonical commutation relations:[ a x , a † x ] = [ a y , a † y ] = [ A ± , A †± ] = 11, all the other commutators being zero. Then, taking Ψ , such that A ± Ψ , = 0 and Ψ n + ,n − := √ n + ! n − ! ( A † + ) n + ( A †− ) n − Ψ , , n ± = 0 , , , . . . , we have N + Ψ n + ,n − = n + Ψ n + ,n − ; N − Ψ n + ,n − = n − Ψ n + ,n − ,H Ψ n + ,n − = ~ ω ( n + + n − + 1) Ψ n + ,n − ,H ↑ Ψ n + ,n − = ~ ω ( n − − n + ) Ψ n + ,n − , which implies that H ↑ Ψ n + ,n − = ~ ω (2 n − + 1) Ψ n + ,n − . If we rather start with a magnetic fieldin the negative z -direction, ~A ↓ = B ( y, − x, H ↓ = H + H ↓ = H − H ↑ , the eigenstatesare again Ψ n + ,n − . In particular we find ( H ↓ Ψ n + ,n − = ~ ω ( n + − n − ) Ψ n + ,n − ,H ↓ Ψ n + ,n − = ~ ω (2 n + + 1) Ψ n + ,n − . We conclude that the eigenvalues of both H ↑ and H ↓ are degenerate (each with infinite degen-eracy), so that what discussed in Section II can be applied. Before going on we also need todefine a map, considered for instance in [11], which works like this: j (Ψ n + ,n − ) = Ψ n − ,n + , for all n + , n − . Hence it is easily seen that j intertwines between H ↑ and H ↓ : j H ↑ = H ↓ j .8 V.1 A first choice of x We take here h := H ↑ = ~ ω (2 N − + 11) and x = A + A − . With this choice N = xx † =( N + + 11)( N − + 11) and [ h , N ] = 0. It is clear that ϕ (1) n := Ψ n , n = ( n + , n − ), is an eigenstateof h and N , with eigenvalues ǫ (1) n − := ~ ω (2 n − + 1) and ν n := ( n + + 1)( n − + 1), respectively.It is clear that ν n > x † ϕ (1) n = 0, for all n .Defining now h = x † h x , N = x † x , and ϕ (2) n = x † ϕ (1) n , and playing a little bit with thecommutation relations, we deduce that h = ~ ω N + N − (2 N − − , ϕ (2) n = p ( n + + 1)( n − + 1) Ψ n − +1 ,n + +1 , N = N + N − , (which is not invertible, by the way!). Defining ǫ (2) n = ǫ (1) n ν n = ~ ω (2 n − + 1)( n + + 1)( n − + 1),it is now easy to check that all our claims are satisfied. For instance h = h † , [ h , N ] = 0, N x = xN , ( h N ) x = xh . The vectors ϕ (2) n satisfy h ϕ (2) n = ǫ (2) n ϕ (2) n and N ϕ (2) n = ν (2) n ϕ (2) n .They are orthogonal but not normalized, in general: D ϕ (2) n , ϕ (2) m E = ν (2) n δ n , m . Moreover, sinceker( x ) = { , Ψ ,n − , Ψ n + , ; n − , n + ≥ } , which is infinite dimensional, the set F is not expectedto be complete in H . Indeed, this is so since we can check that, for instance, the non-zerovector Ψ , is orthogonal to the vectors in F . IV.2 a second choice of x : mixing the quantum numbers As before we take h := H ↑ = ~ ω (2 N − + 11). Now, to discuss the effect of the map j , weconsider x = A + j . Hence N = N + + 11, and [ h , N ] = 0. Further we find h = x † h x = ~ ωN − (2 N + + 11), N = N − and ϕ (2) n = x † ϕ (1) n = √ n + + 1 ϕ (1) n − ,n + +1 . Notice that N is degener-ate as well, but the eigenvalues of ( h , N ) together uniquely fix the eigenvector of the system.As in the previous example, all the properties stated in Section II are recovered explicitly.Moreover, since all the vectors ϕ (1) n + , belong to the kernel of X , F is not complete. This isrelated to the fact that ker( x ) = { } .Notice that the quantum numbers ( n + , n − ) in ϕ (2) n appear to be exchanged with respectto those of ϕ (1) n , and the second number is also changed by one unit. This is, in part, theeffect of the map j and, as discussed in [11], can be related to the appearance of analytic andanti-analytic Hermite polynomials in the analysis of Landau levels.9 Crypto-hermiticity
In this section we will show how losing the self-adjointness of h , rather than being a problem,gives rise to a lot of extra features enriching the structure, at least under suitable conditions.For this reason, we first recall what is meant by crypto-hermiticity of an operator, using thedefinition given in [7]: Definition 2
Let us consider two operators H and Θ acting on the Hilbert space H , with Θ positive and invertible. Let H † be the adjoint of H in H with respect to its scalar product andlet H ‡ = Θ − H † Θ , when this exists. We will say that H is crypto-hermitian with respect to Θ (CHwrt Θ ) if H = H ‡ . Notice that this definition reduces to the standard self-adjointness of H if Θ = 11. Usingstandard facts on functional calculus, the assumptions on Θ imply that the operators Θ ± / are well defined. Hence we can introduce another operator h := Θ / H Θ − / , at least ifthe domains of the operators allow us to do so. More explicitly, h is well defined if, taken f ∈ D (Θ − / ), Θ − / f ∈ D ( H ) and if H Θ − / f ∈ D (Θ / ). Of course, these requirements aresurely satisfied if H and Θ ± / are bounded. Otherwise some care is required. It is easy tocheck that h = h † .The starting point of our analysis is now an operator H which is not self-adjoint but whichis CHwrtΘ, Θ as above. Then H ‡ = Θ − H † Θ = H . Also, we assume that an operator X exists such that, calling N = XX ‡ and N = X ‡ X , we have, first of all, [ H , N ] = 0. Wenotice that, being ‡ an adjoint map, N j = N ‡ j , j = 1 ,
2. In other words, H , N and N are allCHwrtΘ. To simplify the analysis we will work in a single Hilbert space H . All throughout thissection we will assume that the operator h := Θ / H Θ − / is well defined. In particular, thisis so when H , Θ and Θ − are bounded. Then h is self-adjoint, h = h † , and commutes withˆ n := Θ / N Θ − / which is also self-adjoint ˆ n = ˆ n † . Hence we have two commuting, self-adjoint, operators which can be simultaneously diagonalized. Therefore, there exists a familyof vectors F (1) ϕ = { ϕ (1) n,k , ( n, k ) ∈ J } , such that ( h ϕ (1) n,k = ǫ (1) n ϕ (1) n,k , ˆ n ϕ (1) n,k = ν n,k ϕ (1) n,k , (5.1)for all ( n, k ) ∈ J . We see that we are thinking of a possible degeneracy of h , degeneracy whichis lifted by ˆ n . We will assume that F (1) ϕ is an o.n. basis of H and that Θ ± / are bounded.Now, due to our assumptions on Θ, it is clear that ker(Θ ± ) = ker(Θ ± / ) = { } . Hence, calling10 (1) n,k = Θ − / ϕ (1) n,k , the set F (1)Φ = { Φ (1) n,k , ( n, k ) ∈ J } is a Riesz basis of H . It is also clear thatthey are eigenstates of H and N : ( H Φ (1) n,k = ǫ (1) n Φ (1) n,k ,N Φ (1) n,k = ν n,k Φ (1) n,k . (5.2)The frame operator associated to F (1)Φ can be easily computed using the resolution of theidentity for F (1) ϕ : S (1)Φ = P ( k,n ) ∈ J (cid:12)(cid:12)(cid:12) Φ (1) n,k ih Φ (1) n,k (cid:12)(cid:12)(cid:12) = Θ − . It is now very easy to construct a secondRiesz basis, F (1) η = { η (1) n,k , ( n, k ) ∈ J } , which is biorthogonal to F (1)Φ . Its vectors are defined as η (1) n,k = Θ / ϕ (1) n,k = ΘΦ (1) n,k , and we get, as expected, S (1) η = P ( k,n ) ∈ J (cid:12)(cid:12)(cid:12) η (1) n,k ih η (1) n,k (cid:12)(cid:12)(cid:12) = Θ = S (1)Φ − .It is trivial to check that P ( k,n ) ∈ J (cid:12)(cid:12)(cid:12) Φ (1) n,k ih η (1) n,k (cid:12)(cid:12)(cid:12) = 11 and that D Φ (1) n,k , η (1) m,l E = δ n,m δ k,l , as well as ( H † η (1) n,k = ǫ (1) n η (1) n,k ,N † η (1) n,k = ν n,k η (1) n,k . (5.3)These results reflect, essentially, those found in [7]. Here, however, these results are, in a certainsense, doubled . Indeed, extending what we have done in Section II, let us now define a newoperator, H := X ‡ H X , and the new vectors Φ (2) n,k = X ‡ Φ (1) n,k , ( k, n ) ∈ J . It is possible toextend to the present context properties analogous to those in (2.4). In particular we find that H = H ‡ , [ H , N ] = 0 , N X = XN , H N X = XH . (5.4)Moreover, extending again the results of Section II, the set F (2)Φ = { Φ (2) n,k , ( n, k ) ∈ J } is com-plete in H if and only if ker( X ‡† ) = { } , or, equivalently, if ker( X Θ − ) = { } . Under thisrequirement, recalling that the different Φ (2) n,k are also linearly independent, it follows that F (2)Φ is a basis of H , whose frame operator can be written as follows: S (2)Φ = X ( k,n ) ∈ J (cid:12)(cid:12)(cid:12) Φ (2) n,k ih Φ (2) n,k (cid:12)(cid:12)(cid:12) = X ‡ Θ − X ‡† = Θ − / X † Θ X Θ Θ − / , (5.5)where X Θ = Θ / X Θ / . It is possible to check that S (2)Φ admits inverse. This is easily seen if,for simplicity, D ( X ‡† ) = H and if ker( X ) = { } : in this case, for each f ∈ H , D f, S (2)Φ f E = h g, Θ − g i , with g := X ‡† f . Since g = 0 and since ker(Θ) = { } , D f, S (2)Φ f E >
0. Hence S (2)Φ can be inverted and, as a consequence of equation (5.5), also ( X † Θ X Θ ) − exists in H . Moreover,calling ǫ (2) n,k = ǫ (1) n ν n,k ( H Φ (2) n,k = ǫ (2) n,k Φ (2) n,k ,N Φ (2) n,k = ν n,k Φ (2) n,k . (5.6)11epeating here what we did above for the first family of hamiltonians G := ( H , H † , h ),we put h := S (2)Φ − / H S (2)Φ 1 / , ˆ n := S (2)Φ − / N S (2)Φ 1 / and ϕ (2) n,k = S (2)Φ − / ϕ (1) n,k , ( n, k ) ∈ J .The same statements concerning G can now be extended to G := ( H , H † , h ), at leastif ker( X Θ − ) = { } . In this case, among the other properties, we can prove that F (2) ϕ = { ϕ (2) n,k , ( n, k ) ∈ J } is an o.n. basis of H and that h = h † . We can also construct the biorthogonalset F (2) η = { η (2) n,k , ( n, k ) ∈ J } , with η (2) n,k = S (2)Φ − Φ (2) n,k = S (2)Φ − / ϕ (2) n,k , which are eigenstates of H † and N † : ( H † η (2) n,k = ǫ (2) n,k η n,k ,N † η (2) n,k = ν n,k η (2) n,k . (5.7)In analogy with what we have done before, we can further define S (2) η and we get S (2) η = S (2)Φ − .The only difference is that we don’t know if F (2) η and F (2)Φ are Riesz bases or not, since this isrelated to the boundedness of the operators S (2) η and S (2)Φ . As for the intertwining equations,our construction gives rise to many of them. We just list here the following: S (1)Φ H † = H S (1)Φ , and S (2)Φ H † = H S (2)Φ . Other relations involving h , h , S (1) η and S (2) η can be easily deduced.This section show how a rather rich framework can be constructed by just three main ingre-dients: an operator Θ positive and possibly bounded with bounded inverse, a second operator H which is CHwrtΘ, and, last but not least, a third operator, X , such that [ H , XX ‡ ] = 0.While Θ and H are all is needed in the construction of G , X is the main ingredient to moveto G , doubling the results originally deduced for G . We should stress that two interestingfeatures break the symmetry between G and G : the first one is that, while H is degenerate, H is not. This is because its eigenvalues depend on both n and k . The second is a bit moresubtle: in the first part of the construction we move from H and N to the commuting self-adjoint operators h and ˆ n . Since they can be simultaneously diagonalized, we use the set F (1) ϕ of their eigenvectors to construct F (1)Φ and, from this, F (1) η , both being (Riesz) bases of H . Onthe other hand, we give conditions for F (2)Φ to be a basis of H . Then this is automatically aset of eigenstates of H and N , which are used to construct the rest of the structure, and inparticular F (2) ϕ and F (2) η . For instance, as already stated, F (2)Φ and F (2) η are not guaranteed tobe Riesz bases. Remark:–
We end this section recalling that in [7] we have discussed the strong relationsbetween crypto-hermitian operators and non-linear regular pseudo-bosons (NLRPB). This has Notice that in the definitions of G we used Θ − rather than S (1)Φ since they coincide.
12n immediate consequence here: all the results discussed here could be restated for NLRPB aswell. This aspect will not be considered here.
VI Conclusions
In this paper we have discussed a procedure to construct, starting from a self adjoint operator h and a second operator x such that [ h , xx † ] = 0, another operator whose eigenvectors can bededuced from those of h . Some examples arising from bosons, quons and Landau levels havebeen discussed.In the second part of the paper we have extended this construction to crypto-hermitianhamiltonians, showing that the procedure, in this case, can be still settled up and that theresults are doubled . Acknowledgements
The author acknowledges financial support by the Murst.
References [1] F. Cooper, A. Khare and U. Sukhatme,
Supersimmetry and quantum mechanics , WorldScientific, Singapore (2001); G. Junker,
Supersimmetric methods in quantum and statisticalphysics , Springer-Verlag, Berlin Heidelberg (1996)[2] V. Spiridonov,
Exactly solvable potentials and quantum algebras , Phys. Rev. Lett., , 398,(1992); V. Spiridonov, Deformed conformal and supersymmetric quantum mechanics ,Mod.Phys. Lett. A, , 1241, (1992); Fernandez D.J., Rosu H.C., Quantum mechanical spectralengineering by scaling intertwining , Phys. Scripta, , 177-183, (2001)[3] F. Bagarello Extended SUSY quantum mechanics, intertwining operators and coherentstates , Phys. Lett. A, DOI: 10.1016/ j.physleta. 2008.08.047 (2008); F. Bagarello
Vector co-herent states and intertwining operators , J. Phys. A., doi:10.1088/1751-8113/42/7/075302,(2009)[4] F. Bagarello,
Intertwining operators between different Hilbert spaces: connection withframes , J. Math. Phys., DOI: 10.1063/1.3094758 (2009)135] F. Bagarello
Quons, coherent states and intertwining operators , Phys. Lett. A, , 2637-2642 (2009)[6] M. Znojil,
Three-Hilbert-space formulation of Quantum Mechanics SYMMETRY, INTE-GRABILITY and GEOMETRY: METHODS and APPLICATIONS , SIGMA 5 (2009) 001.[7] F. Bagarello, M. Znojil,
Non linear pseudo-bosons versus hidden Hermiticity , J. Phys. A,in press[8] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A.Siviloglou and D. N. Christodoulides,
Observation of PT-Symmetry Breaking in ComplexOptical Potentials , Phys. Rev. Lett. , 093902 (2009);[9] R.N. Mohapatra,
Infinite statistics and a possible small violation of the Pauli principle ,Phys. Lett. B, , 407-411, (1990); D.I. Fivel,
Interpolation between Fermi and Bosestatistics using generalized commutators , Phys. Rev. Lett., , 3361-3364, (1990); Erratum,Phys. Rev. Lett., , 2020, (1992); O.W. Greenberg, Particles with small violations ofFermi or Bose statistics , Phys. Rev. D, , 4111-4120, (1991)[10] F. Bagarello Mathematical aspects of intertwining operators: the role of Riesz bases , J.Phys. A, doi:10.1088/1751-8113/43/17/175203, , 175203 (2010) (12pp)[11] S.T. Ali, F. Bagarello, G. Honnouvo, Modular Structures on Trace Class Operators and Ap-plications to Landau Levels , J. Phys. A, doi:10.1088/1751-8113/43/10/105202,43