Klein four-group and Darboux duality in conformal mechanics

The Klein four-group symmetry of the eigenvalue problem equation for the conformal mechanics model of de Alfaro-Fubini-Furlan (AFF) with coupling constant $g=\nu(\nu+1)\geq -1/4$ undergoes a complete or partial (in the case of half-integer values of $\nu$) breaking at the level of eigenstates of the system. We exploit this breaking of discrete symmetry to construct the dual Darboux transformations which generate the same but spectrally shifted pairs of rationally deformed AFF models for any value of the parameter $\nu$. Two distinct pairs of intertwining operators associated with Darboux duality allow us to construct the complete sets of spectrum generating ladder operators which detect and describe finite-gap structure of each deformed system and generate three distinct species of nonlinearly deformed $\mathfrak{sl}(2,{\mathbb R})$ algebra that give rise to nonlinear deformations of the extended super-conformal structure. We show that at half-integer values of $\nu$, the Jordan states associated with confluent Darboux transformations enter the construction, and the spectrum of rationally deformed AFF systems suffers structural changes.


Introduction
In quantum mechanics, symmetries map the states of a system into its states. If the ground state is invariant under the corresponding transformations, one says that the symmetry is unbroken, otherwise symmetry is (spontaneously) broken. Sometimes, along with a continuous group of symmetry transformations, a discrete symmetry group appears [1], and a nontrivial interplay may occur between both types of symmetries. An interesting case to analyze from this point of view is the conformal mechanics model of de Alfaro, Fubini and Furlan (AFF) [2] 1 with coupling constant ν(ν + 1) ≥ −1/4. Its non-relativistic conformal symmetry and supersymmetric extensions [5,6,7,8,9] find a variety of interesting applications including the particles dynamics in black hole backgrounds [10,11,12,13,14], non-relativistic AdS/CFT correspondence [15,16,17,18], QCD confinement problem [19], physics of Bose-Einstein condensates [20,21], just to name a few. At the same time, the eigenvalue problem equation for the system reveals a discrete Klein four-group symmetry generated by reflection (with respect to ν = 1/2) ν → −ν − 1 of the parameter, and the spatial Wick rotation x → ix accompanied by the change of the eigenvalue's sign E → −E. This symmetry, however, turns out to be completely broken at the level of the quantum states when ν is not a half-integer number : application of the group generators to physical eigenstates produces formal eigenstates which do not satisfy the necessary boundary conditions. In the case of half-integer values of the parameter ν the indicated discrete symmetry breaks partially. In the physics of anyons, where the AFF model is used to generate the transmutation of statistics, half-integer values of ν correspond to the two-particle system of identical fermions [22,23,24]. In the context of the problem we consider here, even though the new solutions with arbitrary value of ν generated by transformations of the discrete group are not acceptable from the physical point of view, they serve to produce new solvable potentials and supersymmetric extensions via the (generalized) Darboux transformations [25,26,27,28,29,30,31].
In this work, with the help of the Klein four-group transformations we address the problem of construction of the Darboux dual schemes to generate rational deformations of the conformal mechanics systems with arbitrary values of the parameter ν. As the dual Darboux schemes produce the same but spectrally shifted pairs of rationally deformed AFF models, their distinct intertwining operators allow us to construct the complete sets of the spectrum generating ladder operators for them and identify the nonlinearly deformed species of conformal sl(2, R) algebra which describe their symmetries. In this way we generalize our earlier results obtained for the case of the AFF model with integer values of ν only [32,33], that were based on the Darboux transformations of the quantum harmonic oscillator. Coherently with the indicated peculiarity of the half-integer values of the parameter ν from the point of view of the Klein four-group transformations, we will see how the Jordan states [34,35,36,37,38,39,40] enter the construction at ν = Z + 1/2 via the confluent Darboux transformations.
The article is organized as follows. In Section 2, we summarize the basic ingredients of the Darboux transformations, and in their context, consider the Jordan states. In Section 3, we present the AFF model, its solutions and symmetries, including the action of the Klein four-group on its eigenstates. The algorithm of construction of the dual Darboux schemes is developed in Section 4. On their basis in Section 5 we assemble the complete sets of the spectrum generating ladder operators, and briefly discuss the deformations of conformal symmetry generated by them. The application of the results is illustrated by an example in Section 6. In Section 7 we summarize the results and discuss some problems to be interesting for further investigation. In two Appendices some technical details are included.

Generalized Darboux transformations
In this section we summarize some properties of the generalized Darboux transformations which will be employed in what follows.

Darboux transformations and intertwining operators
Consider the equation corresponding to the eigenvalue problem of a Schrödinger type operator L 0 . In this section we treat Eq. (2.1) as a formal second order differential equation on some interval (a, b), and in the following sections we take care about its physical nature. Suppose we have a set of solutions ψ k corresponding to eigenvalues λ k , k = 1, . . . , n. We use them as seed states for generalized Darboux transformation and generate the associated eigenvalue problem If the set of the seed states is chosen in such a way that the Wronskian W takes non-zero values on (a, b), then potential of the generated system will also be non-nonsingular there.
In general case, solutions of (2.2) are generated from corresponding solutions of Eq. (2.1), where A n is the differential operator of order n defined recursively as By the construction, ker A n = span{ψ 1 , . . . , ψ n }. Operator A n and its Hermitian conjugate A † n intertwine the operators L 0 and L [n] , and satisfy relations From the first equation in (2.6) one can find that ker A † n = span{A n ψ 1 , . . . , A n ψ n }, where is a linear independent solution of Eq. (2.1) with the same eigenvalue λ, W (ψ λ , ψ λ ) = 1. Similarly to (2.3), A † n Ψ λ = ψ λ for Ψ λ / ∈ ker A † n , and Here and in what follows we consider equalities between wave functions and Wronskians in 'up to a multiplicative constant' sense when the corresponding constant will be inessential. The generalized Darboux transformation possesses the iterative property according to which system (2.2) can be generated alternatively via successive Darboux transformations 2 . This property allows us to get some useful Wronskian identities. Suppose that we have two collections of (formal) eigenstates of (2.1), {φ n } = (φ 1 , . . . , φ n ) and {ϕ l } = (ϕ 1 , . . . , ϕ l ). In the first step, we generate a Darboux transformation by taking the first collection as the set of the seed states, and obtain the intermediate Hamiltonian operator with potential In this way, the states of the second collection {ϕ l } will be mapped into the set of (formal in general case) eigenstates {A n ϕ l } = (A n ϕ 1 , . . . , A n ϕ l ) of the intermediate system, where A n is the corresponding intertwining operator of order n constructed following (2.4). Then, employing these states as the seed states for a second Darboux transformation, we finally obtain a Schrödinger operator with a potential V 2 = V 1 (x) − 2(ln W ({A n ϕ l })) ′′ . Having in mind that the same result will be produced by a onestep generalized Darboux transformation based on the whole set of the chosen eigenstates of the system L 0 , we obtain the equality Consider now the set of two states corresponding to a same eigenvalue λ j , {φ 2 } = (φ 1 = ψ j , φ 2 = ψ j ). In this case W (ψ j , ψ j ) = 1, and the corresponding intertwining operator reduces to A 2 = −(L 0 − λ j ). Using this observation and Eq. (2.8), we derive the equality W (ψ j , ψ j , ϕ 1 , . . . , ϕ l ) = W ({ϕ l }), which is generalized for the relation W (ψ 1 , ψ 1 , . . . , ψ s , ψ s , ϕ 1 , . . . , ϕ l ) = W ({ϕ l }) . (2.9) In the case when functions ϕ 1 , . . . , ϕ l are not obligatorily to be eigenstates of the operator L 0 , the last relation changes for (2.10)

Jordan states
For a given Schrödinger operator L 0 , one can construct a certain set of functions which are not its eigentstates but are annihilated by the action of a certain polynomial of L 0 . Functions of such a nature can be related with the confluent Darboux transformations and are identified as Jordan states [35]. They were used, for example, in the construction of isospectral deformations of the harmonic oscillator systems [39,41], and also they appeared in the context of solutions to the KdV equation [40]. In this subsection we construct Jordan states that are solutions of the fourth order differential equation (L 0 − λ * ) 2 χ * = 0. They will play an important role in subsequent consideration. We employ the following approach: take an eigenstate ψ * corresponding to eigenvalue λ * as a seed state of the Darboux transformation. This provides us with the first order differential operators According to Eq. (2.6), their product gives us the shifted Schrödinger operator A † ψ * A ψ * = L 0 − λ * , whose kernel is spanned by the linear independent states ψ * and ψ * . The problem of constructing Jordan states reduces then to solving equations Their solutions are given, up to a linear combination of ψ * and ψ * , by particular solutions of respective inhomogeneous equations, Here the integration limits are chosen coherently with the region where the operator L 0 is defined, and we have the relations 14) which will be useful to produce nonsingular confluent Darboux transformations. Let us inspect now the role of Jordan states (2.13) in Darboux transformations generated by a set of the seed states {ψ n }. The intertwining operator (2.4) and equations (2.5) and (2.12) give us the relations If the state ψ * (or ψ * ) is annihilated by A n , i.e. if the set of the seed states {ψ n } includes ψ * (or ψ * ), the function A n Ω * (or A nΩ * ) will be an eigenstate of L [n] with eigenvalue λ * which is available to produce another Darboux transformation if we consider L [n] as an intermediate system. Otherwise, the indicated function is a Jordan state of L [n] , and in correspondence with (2.13) we have up to a linear combination with A n ψ * and A n ψ * .
Having in mind that Jordan states appear naturally in the confluent generalized Darboux transformations [35], one can consider directly a generalized Darboux transformation based on the following set of the seed states : (ψ 1 , Ω 1 , . . . , ψ n , Ω n ). This generates a Darboux-transformed system which we denote by L [2n] . The intertwining operator B 2n as a differential operator of order 2n is built according to the same rule (2.4), but with the inclusion of Jordan states into the set of generating functions. By the construction, this operator annihilates the chosen 2n seed states, and one can show that

Solutions and Klein four-group symmetry
The specified above Hamiltonian operator has the physical eigenstates and corresponding energies where L (α) are the generalized Laguerre polynomials, and we do not take care about normalization of wave functions.

Conformal symmetry and ladder operators
Hamiltonian (3.2) is one of the generators of the dynamical symmetry sl(2, R) of the AFF model, which together with the second order differential operators satisfies the commutation relations of the sl(2, R) algebra, Coefficient 4 in the first commutator is the distance between the consecutive energy levels of the AFF system. In correspondence with this, operators C ± ν are the ladder operators. They act on physical states and non-physical eigenstates generated from physical ones by transformations of the K 4 group as follows: When ν > −1/2, the corresponding kernels of C − ν and C + ν are given by In the case of ν = −1/2, the kernel of ladder operators C ± −1/2 are still given by (3.10) but with the states ψ −ν−1,0 and ψ −ν−1,−0 there changed, respectively, for the Jordan states where a, b are constants. When ν does not take half-integer values, up to a redefinition of the integration constants in (2.13), one can also prove the relations For half-integer ν = ℓ − 1/2, by using (2.15) one finds with ℓ = 1, 2, . . . . Acting on the first relation by (C ± ℓ−1/2 ) ℓ , we obtain zero, and conclude that (3.14)

Dual Darboux schemes
If we have two ways to generate the same particular system from the AFF model with a certain value of the parameter ν by employing two distinct Darboux transformations based on distinct sets of the seed states having different behaviour at infinity (being indexes by integers with either plus or minus sign), we say that we have two dual Darboux schemes, or a Darboux duality. This property was discussed and exploited earlier in the case of rational deformations of harmonic oscillator for the construction of the complete sets of the spectrum generating ladder operators as well as for the description of emergent nonlinear extensions of superconformal symmetry appearing in such systems [39,44,41,32]. The purpose of this section is to derive certain Wronskian identities and employ them to construct the dual Darboux schemes. For this, let us choose a generic set of physical and non-physical eigenstates of L ν , where i = 1, . . . , N 1 and j = 1, . . . , N 2 , and, for simplicity, we suppose that |k 1 | < . . . < |k N 1 | and |l 1 | < . . . < |l N 2 |. Remembering that in the case of half-integer ν, ψ ν,n−(ν+1/2) ∝ ψ −ν−1,n , we must take care not to take repeated states. Consider now a scheme of the form (4.1) with non-repeated states, and suppose that both k i and l j carry the same sign for all i and j. Also let us define the index number n N = max (|k 1 |, . . . , |k N 1 |, |l 1 |, . . . , |l N 2 |), which can correspond to a state with index ν or −ν − 1. So if k i and l j carry plus sign, the equality On the contrary, if k i and l j carry minus sign, we have the equality where now r i = n N − |k i | and s j = n N − |l j |. These relations are also valid if one of the numbers N 1 or N 2 is equal to zero, which means that in the corresponding scheme there are only states of the same kind with respect to the first index, −ν − 1 or ν, respectively.
To prove the displayed Wronskian relations, we have to note first the fact that the ladder operators C ± ν can be considered as a pair of the order two intertwining operators obtained via the two dual schemes which do not belong to the schemes described by equations (4.2) and (4.3). Let us consider the schemes (ψ ν,0 , ψ −ν−1,0 ) and (ψ ν,−0 , ψ −ν−1,−0 ) in the case of ν > 1/2, while for ν = −1/2 we take the sets (ψ 1/2,0 , Ω −1/2,0 ) and (ψ 1/2,−0 , Ω −1/2,−0 ). By direct calculation, one finds that these schemes are dual, because they produce the same pair of intertwining operators C ± ν as it is obvious from the structure of their kernels discussed in the previous section. By means of Eq. (2.3), we can write the equalities where z = ±n, n ∈ N. The Wronskian form of these equalities is useful to find the action of the ladder operators on the states ψ r(ν),±0 andΩ −1/2,0 . Using Eqs. (2.9) and (2.19), and equalities it is easy to check that where we see explicitly how the states that diverge (or vanish) at infinity are transformed by the ladder operators into the states that vanish (diverge) in the same limit. Having in mind this, and also using Eqs. (2.8), (2.9) and (2.19), we proceed to prove Wronskian relations (4.2) and (4.3). The following reasoning is valid for ν > 1/2, and in the case of ν = −1/2 we just change ψ −ν−1,±0 and ψ −ν−1,±0 by Ω −1/2,±0 andΩ −1/2,±0 , respectively. Consider Wronskian of the set {α} defined in (4.1) without any repeated state in it (taking into account the proportionality between ψ ν,n−m and ψ −ν−1,n when ν is halfinteger). If the states ψ ν,±0 and ψ −ν−1,±0 do not belong to (4.1), we can replace the Wronskian W ({α}) by where we used relations (2.8), (2.9), (4.5) and (4.6), and {C ∓ ν α} means that the ladder operators act over all the states in the set. On the other hand, if ψ r(ν),±0 belongs to (4.1), we can replace the Wronskian of the initial set of the seed states by where {β 1 } is the scheme {α} with the omitted state ψ r(ν),±0 . Finally, if ψ ν,±0 and ψ −ν−1,±0 belong to (4.1), we have where {β 2 } is the scheme {α} with the states ψ ν,±0 and ψ −ν−1,±0 omitted. Note that in all these three equations we have lowered or increased the index of the states in {α}, and also in the case of Eqs. (4.8) and (4.9) we have included additional states which do not belong to the initial set. Also, we note that an exponential factor has appeared. These identities can be applied to the Wronskian that appears on the right hand side of equations (4.8)-(4.10) which will contribute with a new exponential factor in a new Wronskian, and so on. For this reason, if we restrict the initial set {α} by the conditions described above (that every state in the set has the second index of the same sign), and we repeat this procedure n N + 1 times, with positive (negative) sign of the indexes in (4.8)-(4.10), we finally obtain equation (4.2) or (4.3).
On can also consider in this picture the confluent Darboux transformations. As we will see in the following section, in the context of rational extensions it is important to consider the schemes of the form where n i > m for all i = 1, . . . N. In this case we also can construct a dual scheme, but we have to take into account some peculiarities. In the particular case of m = 0, which implies ν = −1/2, the relations (2.19) and (4.6) play the same role as (2.9) and (4.5) in the first case discused above, and the algortihm of dual schemes can be extended. The generalization for arbitrary values of m is non-trivial, and to do that we have to keep in mind that L ν and L ν+m are related via a simple Darboux transformation. Let us consider the set of the seed states (ψ ν,0 , . . . , ψ ν,m−1 ), whose Wronskian is proportional to x m(2ν+m+1)/2 e ∓mx 2 /2 . The intertwining operators of the associated Darboux transformation satisfy the relation A m L ν = (L ν+m − 2m)A m and its Hermitian conjugate form. From here we obtain that      This kind of diagrams are reading as follows. In the top-line, there appear the ordered states vanishing at infinity, which are ordered from the lowest to the highest second index in wave functions, and which always end in the number without a bar (the first index of wave function is ν). In the bottom-line, there appear the Wick rotated states (second index of wave functions appears with the minus sign), ordered in the same way. The black circles denote the states that appear in the Wronskian arguments in the corresponding dual scheme. The mirror diagrams shown in Fig. 1 correspond to the following Wronskian equalities: whose explicit form is given in Appendix B. The transformation which relates the AFF systems described by L ν with L ν+m can also be understood within this picture. Furthermore, using a diagram similar to those in Fig. 1, one can show that the schemes {∆ + } = (ψ r(ν),0 , . . . , ψ r(ν),m−1 ) and {∆ − } = (ψ r(ν),−0 , . . . , ψ r(ν),−(m−1) ) are dual.

Spectrum generating ladder operators
We apply the Darboux duality property to construct the spectrum generating set of ladder operators and the corresponding non-linear conformal algebras for rational extensions of the AFF model with ν ≥ −1/2. This construction was applied earlier for the case of integer values ν = m in [32], where we took advantage of the dual schemes in the harmonic oscillator system. The construction of C ± ν in terms of the dual schemes for the AFF model itself was also realized in [33]. So the construction we present here generalizes the indicated results.
First we summarize briefly how to construct non-singular rational extensions of the AFF model. Then by the algorithm described above, we obtain the dual schemes and by using the corresponding intertwining operators, we construct the spectrum generating sets of ladder operators for these systems.

Rationally deformed systems
A rational deformation of the AFF model can be generated by taking a set of the seed states {α KA } = (ψ ν,l 1 , ψ ν,l 1 +1 , . . . , ψ ν,lm , ψ ν,lm+1 ) , composed from m pairs of neighbour physical states. Krein-Adler theorem [30,31] guarantees that the resulting system described by the Hamiltonian operator of the form is nonsingular on R + . Here F ν (x) and Q ν (x) are real-valued polynomials, Q ν (x) has no zeroes on R + , its degree is two more than that of F ν (x), and so, the last rational term in (5.2) vanishes at infinity. The spectrum of the system (5.2) is the equidistant spectrum of the AFF model with the removed energy levels corresponding to the seed states. Consequently, any gap in the resulting system has a size 12 + 8k, where k = 0, 1, . . . corresponds to k adjacent pairs in the set (5.1) which produce a given gap. An example of this kind of the systems is generated by the scheme (ψ ν,2 , ψ ν,3 ), whose dual negative scheme is given by equation (4.20).
Another class of rationally extended AFF systems is provided by isospectral deformations generated by the schemes of the form which contain only Wick rotated states ρ 2 (ψ ν,n (x)) = ψ ν,n (ix). As the functions used in this scheme are proportional to x ν+1 and do not have real zeros other than x = 0, one obtains a regular on R + system of the form where f ν (x) is a rational function disappearing at infinity [49], and one can find that potential of the system (5.4) is a convex on R + function. In this case the transformation does not remove or add energy levels, and, consequently, the initial system L ν and the deformed system (5.4) are completely isospectral super-partners. Some concrete examples of the systems (5.4) with integer values of ν were considered in [32]. Consider yet another generalized Darboux scheme which allows us to interpolate between different rationally deformed AFF systems. For this we assume that the initial AFF system is characterized by the parameter ν = µ + m, where −1/2 < µ ≤ 1/2 and m can take any non-negative integer value. For these ranges of values of the parameter ν, real zeros of the functions ψ µ+m,n−m are located between zeros of ψ −(µ+m)−1,n , so that we can rethink the Krein-Adler theorem and consider the scheme where the term 4N is provided by the Gaussian factor in the Wronskian, and the last term is a rational function vanishing at infinity and having no zeros in the whole real line, including the origin, if an only if −1/2 < µ ≤ 1/2, see Appendix A. Let us analyze now some special values of µ The case µ = 0 : by virtue of relation between Laguerre and Hermite polynomials, H 2n (x) = (−4) n n!L n (x 2 ) (−1/2) and H 2n+1 (x) = 2(−4) n n!xL n (x 2 ) (1/2) , in this case we obtain those systems which were generated in [32] by Darboux transformations of the halfharmonic oscillator; they are characterized by gaps of the size 8+4k, and represent rational extensions of the AFF model with integer coupling constant g = m(m + 1), which in the case of m = 0 reduce to a rationally extended harmonic oscillator supplied with a potential barrier at x = 0. Note that the minimal size of the gaps here is less than that for the systems produced by the scheme (5.1).
By considering this last comment, in conclusion we have that when −1/2 ≤ µ < 1/2, the states ψ −(µ+m)−1,n i (and Ω m−1/2,n i −m in the case of µ = −1/2) are non-physical states, which means that only the physical states ψ ν+m,n i −m indicate the energy levels removed under the corresponding Darboux transformation, i.e., there are gaps of the minimum size 2∆E = 8, where ∆E = 4 is the distance between energy levels of the AFF model, which can merge to produce energy gaps of the size 8 + 4k. On the other hand, when µ = 1/2, we have a typical Krein-Adler scheme with gaps of the size 12 + 4k.
To give an example, we put put m = 0, that means ν = µ, and consider the scheme (ψ −ν−1,2 , ψ ν,2 ) with −1/2 < ν ≤ 1/2, whose Wronskian is presented explicitly in B, and in the case of ν = −1/2 we have the scheme (ψ −1/2,2 , Ω −1/2,2 ). The potential of the rationally deformed AFF system generated by the corresponding Darboux transformation is shown in Fig. 2 and Fig. 3. values. For ν = −1/2, the corresponding limit is taken, and the resulting system has an attractive potential with a (not shown) potential barrier at x = 0. For ν = 0, we obtain a rationally extended half-harmonic oscillator. The case ν = 1/2 corresponds to the Krein-Adler scheme (ψ 1/2,1 , ψ 1/2,2 ) with a gap equal to 12. On the right, the corresponding ground states are shown. As it is seen from the figures, the first minimum of the potential grows in its absolute value, its position moves to 0, and it disappears at ν = 1/2, while the local maximum near zero also grows, its position approaches zero, and it goes to infinity in the limit. Besides, the first maximum of the ground state vanishes when ν approximates the limit value 1/2. Coherently with the described behavior of the potential, as it can be checked by a direct computation, the image of the Darboux-transformed state ψ ν,1 , to be the first excited state of the new system when −1/2 ≤ ν < 1/2, vanishes when ν → 1/2, the corresponding energy level disappears from the spectrum at ν = 1/2, and the size of the gap increases from 8 to 12.
The described three selection rules to choose the seed states can be considered as a possible negative scheme (5.3) genenerating isospectral deformations, or as a positive Krein-Adler (5.1) and interpolating (5.5) schemes. Then we can apply the algorithm constructed in Section 4 to obtain the corresponding dual schemes for them. The positive and negative dual schemes will be used in the next subsection to construct complete sets of the spectrum generating ladder operators for the rationally deformed conformal mechanics systems.

Intertwining and ladder operators
As a starting point, we consider any positive scheme for the AFF model L ν that produces its certain non-singular rational deformation. For simplicity we do not touch here the schemes that contain Jordan states. However, we have relations (2.18) and (3.14), and relations (4.2) and (4.3) were extended to such cases with the corresponding substitutions; see the comments for Eq. (4.18). This means that the properties summarized below are also valid for the schemes containing Jordan states.
Let a positive scheme contains n + states labeled by n i , i = 1, . . . , n + , with n n + being the biggest quantum number in the set. We denote by L (+) the system generated by the Darboux transformation based on the set of the chosen seed states. By applying the algorithm from Sec. 4, we obtain the corresponding dual negative scheme with n − = 2n n + + 2 − n + seed states labeled by index −l j with j = 1, . . . , n − and −l n − = −n n + . The resulting system of the Darboux transformation based on the negative scheme we denote by L (−) . By using (4.2) we obtain that the generated Schrödinger operators are mutually shifted for a constant, L (+) − L (−) = ∆E(n n + + 1) = 2(n + + n − ) , ∆E = 4 .

(5.8)
We can construct the corresponding intertwining operators of both schemes by following the rule (2.4). Let us denote by A ± (+) and A ± (−) the intertwining operators of the positive and negative schemes being differential operators of the orders n + and n − , respectively. Some useful properties of these operators are summarized as follows. First, they satisfy the intertwining relations from where one concludes that the operators A − (±) map differently physical eigenstates of L ν as well as non-physical ones obtained from them by action of generators of the K 4 group. The states ψ r(ν),±n behave asymptotically as e ±x 2 /2 , and the states produced from them by application of differential operators A − (±) will carry the same exponential factor. Having this asymptotic behavior in mind, let us suppose that ψ r(ν),−l * and ψ r(ν),n * are some arbitrary states from the negative and positive scheme, respectively. By using (5.9), we obtain the relations in both sides of which the functions satisfy the same second order differential equation and have the same behaviour at infinity. Note that in the dual schemes in (4.2) and (4.3), the indexes n n + − l * and −(n n + − n * ) are in correspondence with the indexes r i , and s i of the states omitted from the positive and negative scheme, respectively. This helps us to provide the identities where the factor (−1) n − is reconstructed by taking into consideration the negative sign in each derivative factor of A + (−) . It is enough to prove the first relation in (5.11), and the second is produced by the Hermitian conjugation. As we know, A − (+) , annihilates all the states in the positive scheme, while A + (−) annihilates all the functions of the form A − (+) ψ r(ν),−l * . Then, acting by A + (−) from the left on both sides of the first relation in (5.10), we find that ker A + (−) A − (+) = (ψ ν,0 , ψ −ν−1,0 , . . . , ψ ν,nn + , ψ −ν−1,nn + ) = ker (C − ν ) nn + +1 . Finally, to have a complete picture we write down the relations With the help of the described intertwining operators, we can construct three types of ladder operators for L (±) which are given by: By using (5.8) and (5.9), we find that they satisfy the commutation relations 14) [L (+) , C ± ] = ±∆E(n n + + 1)C ± , and λ ± i are the corresponding eigenvalues of the seed states in the positive and negative schemes. Equations (5.14) are three different but related copies of the nonlinearly deformed conformal algebra sl(2, R).
In isospectral case, the operators A ± are the spectrum generating ladder operators, where their action on physical eigenstates of L (±) is similar to that of C ± ν in the AFF model. On the other hand, in rationally extended gapped systems obtained by Darboux transfromations based on the schemes not containing Jordan states, the separated states have the form A − (−) ψ −ν−1,−l j = A − (+) ψ ν,n n+ −l j , where the states ψ −ν−1,−l j belong to the negative scheme and ψ ν,n n+ −l j are the omitted states in the corresponding dual positive scheme. Since by construction the separated states belong to the kernel of A + (−) , the operators A ± and C − will always annihilate all them. The resulting total picture related to the action of the ladder operators can be summarized as follows. Operators of the A ± type detect all the separated states organized in valence bands, while they act like ordinary ladder operators in the equidistant part of the spectrum. The lowering operator B − annihilates the lowest state in each valence band, and the raising operator B + annihilates there the highest states, and they also act in an ordinary way in the equidistant part. The operators C ± connect the separated part of the spectrum with its equidistant part, and the lowering operator C − annihilates all the separated states as well as some excited states in the equidistant part according to the rule: if there is no level in the spectrum of energy E n − ∆E(n n + + 1), then the corresponding physical eigenstate of energy E n is annihilated by it. For the case of the confluent Darboux transformations produced on the base of the scheme (4.11) and its dual one, the separated states are A − (−) ψ m− 1 2 ,−l * = A − (+) ψ m− 1 2 ,n N −m−l * , but the picture related to the action of the ladder operators is the same.

Applications: Example
In this section we will apply the machinery of the dual schemes and the construction of nonlinear deformations of the conformal algebra to a non-trivial example of rationally extended system with gaps.

Summary, discussion and outlook
We developed the algorithm for the construction of the dual Darboux schemes for the conformal mechanics model with arbitrary values of the statistics parameter ν and coupling constant g = ν(ν + 1) ≥ −1/4. The physical eigenstates together with non-physical ones generated from them by transformations of the Klein four-group form a base for the Darboux duality in the case of ν = Z+1/2. In the case of half-integer values of ν the Jordan states naturally enter the construction via the confluent Darboux transformations. Each pair of the dual Darboux transformations generates a rational deformation of the AFF model which can be completely isospectral to the initial system (up to a global spectral shift), or may have finite number of valence bands in the low part of the spectrum, which are separated by gaps between themselves and from the semi-infinite band with equidistant energy levels. The minimal size of a gap in our construction corresponds to one missing energy level in comparison with two missing levels in gaps of minimal size in the systems generated by the Krein-Adler transfromations, which also are included in our dual Darboux schemes. We showed that when the statistics parameter varies continuously, the spectrum of rationally deformed AFF systems suffers structural changes at half-integer ("fermionic") values of ν. No such changes happen, however, at integer values of ν corresponding to the case of bosons in the context of the statistics transmutations [22,23,24]. Recall that all the deformations of the conformal mechanics model with ν ∈ Z can be generated by generalized Darboux transformations from the quantum harmonic oscillator system [32]. At the same time we also note here that the question of the self-adjoint extension of the AFF Hamiltonian operator requires a special consideration in the case ν = −1/2 [46], which corresponds to a minimal value of the coupling constant g for which the spectrum is bounded from below, and when, as we saw, the Klein four-group symmetry suffers minimal breaking.
The Darboux duality allowed us to obtain the set of the three pairs of ladder operators of different but complementary nature. In the case of the rationally deformed gapped systems, Hermitian conjugate ladder operators of the A type detect all the separated states, each of which is annihilated by both, the lowering and the raising, operators; the lowering operator A − also annihilates the lowest state in the equidistant part of the spectrum. The raising ladder operator of the B type detects the states with highest energy level in each valence band by annihilating them. The lowering operator B − makes the same with the states of the lowest energy level in each valence band, and also annihilates the lowest state in the equidistant part of the spectrum. Although the operators of these two types detect all the separated states as well as identify the borders of the valence bands and the edge of the semi-infinite band with equidistant energy levels, they cannot connect the states from different bands. This job is realized with the help of the ladder operators of the C type. As a result, one can see that any of the two sets of the ladder operators, (C ± , A ± ) or (C ± , B ± ), forms the complete set of the spectrum generating ladder operators by which any eigenstate of the rationally deformed AFF system can be transformed into its any other eigenstate. In the case if we have an isospectral deformation of the AFF system L ν obtained via the Darboux scheme (5.3), the operators A ± are enough to generate the entire tower of physical eigenstates starting from any physical eigenstate.
Each of the three pairs of the conjugate ladder operators together with the Hamiltonian operator generate some nonlinearly deformed version of the conformal sl(2, R) algebra of the W type [50], which is the symmetry of the corresponding rationally deformed AFF system. We, however, did not compute commutators between ladder operators of different types. Following ref. [33], where the AFF systems with ν ∈ Z and their rational deformations were considered in detail, one can show that the indicated commutators give rise to the sets of operators of the form where index k can take integer values from 1 to infinity. A more careful inspection of the nature of these operators shows, however, that they are reduced to the products of the extended but finite family of the intertwining operators of the form A ± (±) (C ± ν ) k and A ± (±) (C ∓ ν ) k , and their Hermitian conjugate ones. Though the resulting picture is expected to be rather complicated and requires a separate study, it should be similar to that appearing in the case of ν = 0, which was analyzed in detail in [33], as well as to that in the P Tregularized two-particle Calogero systems [51,52], and can be summarized as follows. Any extended system composed from a pair of the AFF systems characterized by parameters ν and ν + m, m ∈ Z, or from a pair of rationally deformed AFF systems with the same fractional part of their corresponding parameters ν is described by a nonlinearly extended superconformal osp(2|2) algebra, whose two-by-two matrix generators are constructed from the extended family of the intertwining operators mentioned above.
Our consideration of rational deformations of the conformal mechanics was restricted by inclusion of Jordan states of the simplest form. Following the analysis and ideas presented in refs. [34,35,36,37,38], the constructions can be generalized to the case of higher order Jordan states defined via relations (L − λ * )Ω (0) * = ψ * , (L − λ * )Ω (k) * = Ω (k−1) * , k = 1, . . . , (7.4) as well as to their further generalizations defined as the states annihilated by polynomial in L operators [39]. The states that we have used correspond to Ω (0) * . In this way, one can produce the systems by means of confluent Darboux transformations which involve more Jordan states of these chains, and one can expect that the spectrum of the resulting systems will have a similar gapped structure. Then it would be interesting to study such kind of the systems from the point of view of the spectrum generating ladder operators and the extended nonlinear deformations of the (super)conformal algebra associated with them.
It is known that the conformal symmetry underlies the relation between the quantum free particle and harmonic oscillator systems [53,15]. A similar picture also is valid for the two-particle Calogero system (without confining potential term and omitted center of mass degree of freedom) and the AFF model [2,19,54,55]. The Calogero model and its deformations, in turn, are intimately related to the soliton solutions of the Korteweg-de Vries equation and higher equations of its hierarchy [56,57,58,40]. It would be interesting to investigate the question of a possible relation between rational deformations of the AFF model studied here and solutions to the same hierarchy of completely integrable systems described by partial differential equations. At the same time we note that the approach based on the Klein four-group transformations employed here can also be applied to the two-particle Calogero model with arbitrary values of the statistics parameter ν but without the confining harmonic potential term. In this way one could expect to generate new quantum solvable systems which may be related to the Korteweg-de Vries hierarchy.
In the case µ = −1/2, we put ϕ j = ψ m−1/2,n j+1 −n with j = 0, . . . , l − 1, and then we arrive at the relations This relation is true for the case i = 1, which implies that A 2 → B 2 in the corresponding limit. The general case is proved by induction.