Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials

We construct rational extensions of the Darboux-P\"oschl-Teller and isotonic potentials via two-step confluent Darboux transformations. The former are strictly isospectral to the initial potential, whereas the latter are only quasi-isospectral. Both are associated to new families of orthogonal polynomials, which, in the first case, depend on a continuous parameter. We also prove that these extended potentials possess an enlarged shape invariance property.


Introduction
Since the seminal paper of Gómez-Ullate, Kamran, and Milson [19], which introduced the concept of Exceptional Orthogonal Polynomials (EOP), the discovery of their connection with translationally shape invariant quantum potentials (TSIP) by Quesne [34,38], and the construction of infinite sets of such potentials by Odake and Sasaki [33], much progress has been made in the understanding of exactly solvable systems related to orthogonal polynomials (see [18] and references therein). The key tool to generate such systems is the Darboux or Darboux-Bäcklund transformation (DBT), which connects pairs of intertwined Hamiltonians. Starting from one primary TSIP, specific symmetries of this last select the quasi-polynomial formal eigenfunctions that can be used as seed functions to build chains of rationally extended potentials. The eigenstates of these extensions are then (up to a gauge factor) Exceptional Orthogonal Polynomials, which, by using Crum formulas, can be expressed as Wronskians of classical orthogonal polynomials. The regularity properties of the chains, including degenerate chains (i.e., chains with repeated use of the same seed functions), are controlled by enlarged versions of the Krein-Adler theorem [2,14,18,28,39]. For some chains, the extended potentials share the same shape invariance properties as the primary potential [23,25,35,38]. With other choices of seed functions, the resulting potentials possess an enlarged shape invariance property [24,36,37].
Until now, the chains of extensions were "rigid" in the sense that they were uniquely determined by the tuple of associated seed functions. Very recently, with B. Bagchi [3], we obtained new rational extensions of the Darboux-Pöschl-Teller potential (based on the so-called para-Jacobi polynomials [7]), which depend on a free parameter and can then be modulated continuously. The eigenstates of these extended potentials are associated to new families of orthogonal polynomials that are, in a broad sense, exceptional para-Jacobi polynomials and which depend on a free continuous parameter.
In this paper, we consider the possibility of building new rational extensions of two confining TSIP, namely the trigonometric Darboux-Pöschl-Teller (TDPT) and isotonic potentials, via confluent chains of DBT, that is chains of DBT in which the spectral parameters of the different seed functions converge to the same value. It has to be noticed that it is precisely by using such confluent chains applied to the constant potential and considering the associated rational extensions that Adler and Moser built the Burchnall-Chaundy polynomials [6] in their seminal paper on the rational solutions of the KdV equation [1].
After recalling in Section 2 the basic elements concerning the Darboux-Bäcklund transformations, we introduce in Section 3 the concept of confluent chains of DBT. We show in particular that the confluent chains of arbitrary order can be generated within the standard frame of (completed) DBT chains, giving rise to multiparameter dependent extensions in a way similar to that used by Keung et al. to produce n-parameter families of isospectral potentials [10,27].
In Section 4, applying two-step confluent chains of DBT for which the seed functions are eigenstates, we build regular rational extensions of the TDPT with appropriate parameters. The extended potentials depend on a continuous parameter and are strictly isospectral to the initial potential. The eigenstates form new families of orthogonal polynomials, which have a free parameter dependence. We exhibit particular examples and prove that the extended potentials present an enlarged shape invariance property, in which the parameter transformation acts in a non trivial way on the supplementary parameter.
In Section 5, we make the same construction for the isotonic system. In contrast with the TDPT case, the regular rational extensions do not depend on any supplementary degree of freedom and we only have quasi-isospectrality between the extended potentials and the original one. We also furnish explicit examples of extensions and establish their enlarged shape invariance property.
Finally, Section 6 contains the conclusion.

Darboux-Bäcklund transformations: basic elements
We consider a one-dimensional Hamiltonian H = −d 2 /dx 2 +V (x), x ∈ I ⊂ R, and the associated Schrödinger equation ψ λ (x) being a formal eigenfunction of H for the eigenvalue E λ . In the following, we suppose that, with Dirichlet boundary conditions on I, H admits a discrete spectrum of energies and eigenstates (E n , ψ n ) n∈{0,...,nmax}⊆N , where, without loss of generality, we can always suppose that the ground level of H is at zero From any solution ψ ν (or equivalently w ν ), we can build a Darboux-Bäcklund transformation (DBT) A (w ν ) defined as [8,9,12,13,22]    w λ (x) where A (w ν ) is a first-order differential operator given by λ are respectively solutions of the Schrödinger and RS equations with the same energy E λ as in Eqs. (2.1) and ( 2.2), but with a modified potential that we call an extension of V (x). For the associated Hamiltonian we have the intertwining relations The function ψ 3) can then be rewritten as the Darboux-Crum formula where W (y 1 , ..., y m | x) denotes the Wronskian of the family of functions y 1 , ..., y m , The eigenfunction ψ ν is called the seed function of the DBT A(w ν ) and V (ν) and ψ (ν) λ are the Darboux transforms of V and ψ λ , respectively.
Note that A(w ν ) annihilates ψ ν and, consequently, Eqs. (2.3) and (2.5) allow to obtain an eigenfunction of V (ν) for the eigenvalue E λ only when λ = ν. Nevertheless, we can readily verify that 1/ψ ν (x) is such an eigenfunction. By extension, we then define the "image" by A(w ν ) of the seed eigenfunction ψ ν itself as .
At the formal level, the DBT can be straightforwardly iterated and a chain of m DBT can be simply described by the following scheme where N j denotes the j-uple (ν 1 , ..., ν j ) (with N 1 = ν 1 ), which completely characterizes the chain. We denote by (N m , ν m+1 , ..., ν m+k ) the chain obtained by adding to the chain N m the DBT associated to the successive eigenfunctions ψ is an eigenfunction associated to the eigenvalue E λ of the potential (see Eq. (2.4)) and can be written as (cf. Eqs. (2.3) and (2.5)) A chain is non-degenerate if all the spectral indices ν i of the chain N m are distinct and is degenerate if some of them are repeated in the chain. For non-degenerate chains, Crum has derived very useful formulas for the extended potentials and their eigenfunctions in terms of Wronskians of eigenfunctions of the initial potential [11].
Crum's formulas When all the ν j and λ are distinct, we have and

Confluent chains of DBT
The single-confluent limit of a chain of DBT N m is obtained when all the spectral indices ν j tend simultaneously to the same value ν j → ν, ∀j ∈ {1, ..., m} (in the following, we consider only single-confluent chains and than omit the adjective "single").

Two-step confluent chains
We consider a chain of two DBT N 2 = (ν 1 , ν 2 ), which, in the non-degenerate case ν 1 = ν 2 , gives (see Eqs. (2.7), (2.8), (2.9), and (2.10)) (3.1) Note that in the degenerate case ν 1 = ν 2 , we have (see Eq. (2.6)) ψ (ν 1 ) and by applying the DBT A(w (ν 1 ) , we recover simply the initial potential V , . The confluent case corresponds to the limit ν 2 → ν 1 . As proven by Fernández et al. [5,17], the confluent extended potential and its eigenstates admit the following integral representations Both depend on an arbitrary real parameter W 0 and for an adapted range of W 0 values, the extended potential is regular. In fact, the formula for the potential (3.2) already appears in 1986 in a paper of Luban and Pursey [29] and a few years later in [27]. The Matveev formulas [20,30,31] for the two-step case which express the confluent extension and its eigenstates in terms of generalized Wronskians (which are in fact two-way, or double Wronskians [42]) can be viewed as associated to a particular choice of the W 0 constant. Indeed, if we consider the indexed family of RS functions w ν (x) as satisfying a prescribed initial condition in x 0 , we have, in the confluent limit, w ν 2 (x) → w ν 1 (x). It results from Eq. (3.1) that But we can readily verify that and since in this case we also have [32] ∂w ν (x) ∂E ν we see that Eq. It has to be noticed that the degenerate extension V (ν 1 ,ν 1 ) (x) can be recovered from the confluent one V (ν 1 ,ν 1 ) (x) by taking the singular limit value W 0 → ∞.
Fernández et al. used these formulas to generate new second-order SUSY partners of the free particle, the Kepler-Coulomb, and the single-gap Lamé potentials.
The preceding results can be in fact integrated within the standard DBT scheme simply using the DBT in its completed form (see Eq. (2.6)) as in [10,27]. Indeed, by applying the DBT A(w ν ), we generate first the one-step (possibly singular) extension for the eigenvalue E ν , the most general eigenfunction (up to a multiplicative factor) of V (ν) for the same eigenvalue is , λ 1 ∈ R, (3.5) and the corresponding RS function is .
We can now use this general solution as seed function for the second DBT. Then applying A(W (ν) ν ) to V (ν) , we obtain the following second extension and we recover the first Fernández formula (3.2) with λ 1 = −W 0 . As for the eigenfunctions of V (ν,ν) , they are given by (µ = ν) . (3.7)

General multi-step confluent chains
Fernández and Salinas-Hernández [16] have also considered the so-called "hyperconfluent" case corresponding to a three-step confluent DBT, for which they have extended the previous formulas, Eqs. (3.2) and (3.3). For these three-step extensions, the potential depends on two arbitrary real parameters. In fact, the preceding analysis allows to obtain integral formulas "à la Fernández" for chains of arbitrary order in a very simple way. In the following, the symbol ν l means (ν, ..., ν) l times .
In the three-step case, the image of Ψ where we have used the notation Λ m = (λ 1 , ..., λ m ). The next extension generated by the DBT We recover the "hyperconfluent" third-order superpartner of V (x) as obtained by Fernández and Salinas-Hernández [16]. Within this scheme, the generalization is immediate and repeating the procedure m times, we obtain for the hyperconfluent m th -order extension of V (x) the expression with the following recurrence relation for the successive seed functions . In other words, whenever m is even (m = 2k) and whenever m is odd (m = 2k + 1) The eigenstates of V (ν m ) can be obtained by successive applications of the A W (ν l ) ν operators (the product being ordered in decreasing order), A direct application of the Crum Wronskian formulas [11] to these general, parameterdependent, confluent extensions is obviously not possible and, as mentioned above, the Matveev formulas [20,30,31] correspond only to a particular choice of the λ j parameters. Nevertheless, these extended potentials are amenable to other (standard) Wronskian formulas [5,17,40].
In the following, we limit our analysis to the case of two-step DBT. We are interested in the possibility of building regular and rational extensions with such confluent chains, which turns out to be possible for the trigonometric Darboux-Pöschl-Teller (TDPT) potential and the isotonic potential.

General scheme
The trigonometric Darboux-Pöschl-Teller (TDPT) potential (with zero ground-state energy) is defined on with α, β > 1/2. Its physical spectrum, associated to the asymptotic Dirichlet boundary conditions is given in terms of Jacobi polynomials [15,21,41] In the following we consider the case where α and β are integers: If we choose as initial seed function an eigenstate ψ n (x; N, M ) of V (x; N, M ), by taking is a polynomial of degree N + M + 2n + 1 in z with [15,41] Note the following recurrence From Eq. (3.6), we then obtain for the confluent two-step extension V (n 2 ) , In this case, its eigenfunctions for k = n are given by (see Eqs. (3.5) and (3.7)) is an exceptional Jacobi polynomial in the broad sense of the term. Hence, for k = n, The orthogonality conditions between eigenstates imply that the P (n 2 ) N,M,k (z; λ 1 ) constitute a family of orthogonal polynomials (indexed by k ∈ N) on ]−1, 1[ with respect to the measure

Shape invariance of
Consequently, in the n = 0 case, we have and we obtain an enlarged shape invariance property The preceding condition (4.4) can be rewritten as

Equation (4.6) is equivalent to
where d dz A (z) is given by As for d dz B(z), using the derivation formula [15,41] it can be expressed as But the differential equation satisfied by the Jacobi polynomials is [15,41] 1 − z 2 d 2 dz 2 P ( Consequently 4n To satisfy the equality (4.7), we then must choose In this case, Eq. (4.6) simply becomes In the limit z → 1 − , we obviously get and (see Eqs. (4.1) and (4.2)) We conclude that for n ≥ 1, the two-step confluent rational extensions of the TDPT potential satisfy the enlarged shape invariance property (4.5) with λ ′ 1 given in Eq. (4.9). Note that the latter relation ensures that the domain of λ 1 values for which V (n 2 ) (x; N, M, λ 1 ) is regular corresponds exactly to the domain of λ ′ 1 values for which V ((n−1) 2 ) (x; N + 1, M + 1, λ ′ 1 ) is also regular (see Eqs. (4.2) and (4.3)).
In the n = N = M = 1 case, such a property can be directly verified as follows. We get

as well as
Hence the identity 5 Two-step confluent rational extensions of the isotonic potential

General scheme
The isotonic oscillator potential (with zero ground-state energy E 0 = 0) is defined on the positive half-line ]0, +∞[ by If we add Dirichlet boundary conditions at zero and infinity and if we suppose α > 1/2, it has the following spectrum (z = ωx 2 /2) is the usual Laguerre polynomial [15,41]. In the following α is an integer: α = N . If we choose as initial seed function an (unnormalized) eigenstate ψ n (x; ω, N ) of V (x; ω, N ), by taking x 0 = 0 we get is a polynomial of degree N + 2n.
In contrast with what happened in the TDPT case, to obtain a rational extension we need now to eliminate the exponential factor, which necessitates fixing λ 1 = − ψ n 2 . Equation (3.6) then becomes where V (n 2 ) (x; ω, N, − ψ n 2 ) constitutes a rational extension of V (x; ω, N ). Since F N n (z) is a negative, strictly increasing function of z, such that lim x→+∞ F N n (z) = 0, Q N n (z) keeps a constant strictly negative sign, i.e., Q N n (z) has no zero on the positive half-line. This shows that V (n 2 ) (x; ω, N, − ψ n 2 ) is also regular. Let us note too that [15,41] and , that is, k (x; ω, N, − ψ n 2 ) decreases exponentially to zero at infinity and also tends to zero at the origin (see Eq.(5.5)). It is therefore an admissible eigenstate of V (n 2 ) (x; ω, N, − ψ n 2 ) for every k = n.
Note that in this case is not an eigenstate since it is not normalizable. This implies that V (n 2 ) (x; ω, N, − ψ n 2 ) and V (x; ω, N ) are only quasi-isospectral, the two-step confluent DBT A(W The orthogonality conditions between eigenstates imply that the set of L (n 2 ) k,N (z) with k = n, constitute a family of orthogonal polynomials, indexed by k ∈ N, on the positive half-line with respect to the mesure

The n = 0 case
The n = 0 case gives no new extended potentials with respect to previously known results. Indeed, the corresponding confluent two-step DBT is obtained by applying successively the onestep DBT A(w 0 ) and A(W (0) 0 ). The first one is the usual SUSY partnership and, due to the shape invariance property of the isotonic potential, we have where V (0) (x; ω, N ) admits the following eigenstates for the respective energies E k (ω), k ≥ 1.
The second DBT A W (0) 0 is associated to the seed function Ψ By taking x 0 = 0 and λ 1 = − ψ 0 2 , we obtain (5.8) where is a formal eigenfunction of V (0) (x; ω, N ) = V (x; ω, N +1)+2ω that tends to zero at infinity and diverges at the origin. Such an eigenfunction is unique (up to a constant multiplicative factor) at a given eigenvalue (here E 0 = 0) [4,26].
Consequently Ψ 0 (x; ω, N, − ψ 0 2 ) coincides necessarily (up to a constant multiplicative factor) with a type II seed function [18] of V (x; ω, N + 1) + 2ω, which is a formal eigenfunction for the eigenvalue E −1 + 2ω, namely Note that this result can be recovered directly by using the identity (see Eq. which coincides (up to a constant shift) with the usual extensions obtained with type II seed functions [18]. Explicitly, we get In the N = 1 case, this gives

The n = 1 case
We have in this case The eigenstates of V (1 2 ) are (k = 1) , that is, In particular, for N = 1 Moreover (k = 1) The polynomial is associated to the ground state of V (1 2 ) (x; ω, 1, − ψ 1 2 ) and which has a single zero at √ 2 on ]0, +∞[, corresponds to the first excited eigenstate.

Shape invariance of the two-step confluent rational extensions of the isotonic potential
V (x; ω, N ) is a translationally shape invariant potential with a SUSY partner and (the coefficient is readily established by a direct calculation) Considering the SUSY partner of V (n 2 ) (x; ω, N, − ψ n 2 ), let us deal separately with the n ≥ 1 and n = 0 cases. For n ≥ 1, this partner is given by Inserting (5.12) into (5.11), we get More precisely, on using (5.6), If, for an appropriate constant C, the following condition is satisfied, then and we obtain an enlarged shape invariance property 14) The preceding condition (5.13) can be rewritten as and Equation ( As for d dz B(z), using the derivation formula [15,41] it can be expressed as (5.18) The differential equation satisfied by the Laguerre polynomials is [15,41]  Hence, to satisfy the first condition (5.16), we must choose With such a choice for C, the second condition (5.17) is then also fulfilled because it is obvious that B(0) = 0 and, from Eq. (5.4), it results that We conclude that for any n ≥ 1, the two-step confluent extension V (n 2 ) SUSY (x; ω, N, − ψ n 2 ) satisfies the enlarged shape invariant property (5.14).
Let us now turn ourselves to the n = 0 case. As noted above (see Eq. (5.7)), the formal eigenfunction However, it can be readily seen that Eq. (5.19) cannot be satisfied by any C, so that we conclude that we do not have any strict nor enlarged shape invariance for the confluent extension V (0 2 ) .

Conclusion
Using two-step confluent chains of DBT, we have generated new families of orthogonal polynomials, associated to novel regular rational extensions of the isotonic and TDPT potentials, exhibiting an enlarged shape invariance property. Interestingly, in the second case, the orthogonal polynomials depend on a free parameter that can be modulated continuously, a feature already encountered for other extensions based on para-Jacobi polynomials. Considering chains of arbitrary order and the possibility of obtaining more general families of orthogonal polynomials subjected to multi-parameter dependence would be a very interesting topic for future investigation.