Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 11 (2015), 061, 26 pages
Conf luent Chains of DBT: Enlarged Shape Invariance
and New Orthogonal Polynomials?
Yves GRANDATI
†
and Christiane QUESNE
‡
†
Equipe BioPhysStat, LCP A2MC, Université de Lorraine-Site de Metz,
1 bvd D.F. Arago, F-57070, Metz, France
E-mail: grandati@yahoo.fr
‡
Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles,
Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium
E-mail: cquesne@ulb.ac.be
Received March 26, 2015, in final form July 15, 2015; Published online July 28, 2015
http://dx.doi.org/10.3842/SIGMA.2015.061
Abstract. We construct rational extensions of the Darboux–Pöschl–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.
Key words: quantum mechanics; supersymmetry; orthogonal polynomials
2010 Mathematics Subject Classification: 81Q05; 81Q60; 42C05
1
Introduction
Since the seminal paper of Gómez-Ullate, Kamran, and Milson [23], which introduced the concept
of exceptional orthogonal polynomials (EOP), the discovery of their connection with translationally shape invariant quantum potentials (TSIP) by Quesne [41, 42], and the construction of
infinite sets of such potentials by Odake and Sasaki [39], much progress has been made in the
understanding of exactly solvable systems related to orthogonal polynomials (see [22] 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 [3, 17, 22, 32, 47]. For some chains, the extended potentials share the same shape invariance properties as the primary potential [24, 25, 42, 43]. With other choices of seed functions,
the resulting potentials possess an enlarged shape invariance property [26, 44, 45].
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 [4], we obtained new rational extensions of the Darboux–Pöschl–Teller potential (based on the so-called
para-Jacobi polynomials [9]), which depend on a free parameter and can then be modulated
?
This paper is a contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of
Luc Vinet. The full collection is available at http://www.emis.de/journals/SIGMA/ESSA2014.html
2
Y. Grandati and C. Quesne
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 [8] in their seminal
paper on the rational solutions of the KdV equation [2].
The possibility of considering two successive factorization energies tending towards a common
real value in a chain of Darboux transformations was first considered in the framework of phaseequivalent potential construction [5], then extended to a class of potentials defined on the
line [49]. These approaches generalized to arbitrary bound-state energies a procedure already
known for the ground state and the first few excited states [13, 31]. An independent proposal
was made wherein the terminology “confluent” algorithm was introduced and some partners
of the free particle and the harmonic oscillator were exhibited [37]. The confluent algorithm
was then studied in more general terms and applied to the free particle, one-soliton well, and
harmonic oscillator [19]. It was also considered in a general construction of all possible first- and
second-order partners of the TDPT potential [12]. The “hyperconfluent” third-order algorithm,
wherein the three factorization energies converge to the same value, was analyzed [21]. Some
Wronskian formulas applicable to the confluent case were also derived [7, 20, 48].
The present work differs from the previous ones devoted to confluent chains by the restriction
to final potentials that are rational extensions of the initial ones. As a consequence of this
condition, some parameters of the latter may have to be chosen integer.
After recalling in Section 2 the basic elements concerning the Darboux–Bäcklund transformations, we review 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 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 nontrivial 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. Section 6 contains some final comments.
2
Darboux–Bäcklund transformations: basic elements
b = −d2 /dx2 +V (x), x ∈ I ⊂ R, and the associated
We consider a one-dimensional Hamiltonian H
Schrödinger equation
ψλ00 (x) + (Eλ − V (x))ψλ (x) = 0,
(2.1)
b for the eigenvalue Eλ . In the following, we suppose
ψλ (x) being a formal eigenfunction of H
b admits a discrete spectrum of energies and
that, with Dirichlet boundary conditions on I, H
Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
3
eigenstates (En , ψn )n∈{0,...,nmax }⊆N , where, without loss of generality, we can always suppose that
b is at zero (E0 = 0).
the ground level of H
The Riccati–Schrödinger (RS) function wλ (x) = −ψλ0 (x)/ψλ (x) associated to ψλ satisfies the
corresponding Riccati–Schrödinger equation [28]
−wλ0 (x) + wλ2 (x) = V (x) − Eλ .
(2.2)
From any solution ψν (or equivalently wν ), we can build a Darboux–Bäcklund transformation
(DBT) A(wν ) defined as [10, 11, 15, 16, 28]
A(wν )
(ν)
wλ (x) → wλ (x) = −wν (x) + (Eλ − Eν )/(wν (x) − wλ (x)),
Z
A(wν ) (ν)
(ν)
b ν )ψλ (x),
ψλ (x) → ψλ (x) = exp − dxwλ (x) ∼ A(w
λ 6= ν,
(2.3)
b ν ) is a first-order differential operator given by
where A(w
b ν ) = d/dx + wν (x).
A(w
(ν)
(ν)
ψλ and wλ are respectively solutions of the Schrödinger and RS equations with the same
energy Eλ as in equations (2.1) and (2.2), but with a modified potential
V (ν) (x) = V (x) + 2wν0 (x),
(2.4)
b (ν) = −d2 /dx2 + V (ν) (x),
that we call an extension of V (x). For the associated Hamiltonian H
we have the factorizations
b (ν) = A(w
b ν )A
b+ (wν ) + Eν ,
H
b =A
b+ (wν )A(w
b ν ) + Eν ,
H
with
b+ (wν )ψ (ν) (x).
ψλ (x) ∼ A
λ
(ν)
The function ψλ in equation (2.3) can then be rewritten as the Darboux–Crum formula
(ν)
ψλ (x) ∼
W (ψν , ψλ | x)
,
ψν (x)
(2.5)
where W (y1 , . . . , ym | x) denotes the Wronskian of the family of functions y1 , . . . , ym ,
y1 (x)
.
.
.
y
(x)
m
.
...
W (y1 , . . . , ym | x) = . . .
(m−1)
(m−1)
y
(x) . . . ym
(x)
1
(ν)
The eigenfunction ψν is called the seed function of the DBT A(wν ) and V (ν) and ψλ are
the Darboux transforms of V and ψλ , respectively.
Note that A(wν ) annihilates ψν and, consequently, equations (2.3) and (2.5) allow to obtain
an eigenfunction of V (ν) for the eigenvalue Eλ only when λ 6= ν. Nevertheless, we can readily
verify that 1/ψν (x) is such an eigenfunction. By extension, we then define the “image” by A(wν )
of the seed eigenfunction ψν itself as
ψν(ν) (x) ∼
1
.
ψν (x)
(2.6)
4
Y. Grandati and C. Quesne
At the formal level, the DBT can be straightforwardly iterated and a chain of m DBT can
be simply described by the following scheme
(Nm−1 )
(N )
ψλ
V
A(wν1 )
(ν )
ψλ 1
A(wν1 )
V (ν1 )
A wν2 1
(N2 )
ψλ
A
(N )
wν2 1
···
V (N2 ) · · ·
A wνm
(Nm )
ψλ
A
(N
)
wνmm−1
,
V (Nm ) ,
where Nj denotes the j-uple (ν1 , . . . , νj ) (with N1 = ν1 ), which completely characterizes the
chain. We denote by (Nm , νm+1 , . . . , νm+k ) the chain obtained by adding to the chain Nm the
(Nm+k−1 )
(Nm )
.
DBT associated to the successive eigenfunctions ψνm+1
, . . . , ψνm+k
(Nm )
ψλ
V
is an eigenfunction associated to the eigenvalue Eλ of the potential (see equation (2.4))
(Nm )
(x) = V (x) + 2
m
X
0
(N
)
wνj j−1 (x)
(2.7)
j=1
and can be written as (cf. equations (2.3) and (2.5))
(Nm )
ψλ
b wν(Nm−1 ) ψ (Nm−1 ) (x) = A
b wν(Nm−1 ) · · · A(w
b ν )ψλ (x).
(x) = A
1
λ
m
m
(2.8)
A chain is non-degenerate if all the spectral indices νi of the chain Nm 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 [14].
Crum’s formulas. When all the νj and λ are distinct, we have
(Nm )
ψλ
(x) =
W (Nm ,λ) (x)
W (Nm ) (x)
(2.9)
and
00
V (Nm ) (x) = V (x) − 2 log W (Nm ) (x) ,
(2.10)
where W (Nm ) (x) = W (ψν1 , . . . , ψνm | x).
3
Conf luent chains of DBT
The single-confluent limit of a chain of DBT Nm is obtained when all the spectral indices νj
tend simultaneously to the same value νj → ν, ∀ j ∈ {1, . . . , m} (in the following, we consider
only single-confluent chains and than omit the adjective “single”).
3.1
Two-step conf luent chains
We consider a chain of two DBT N2 = (ν1 , ν2 ), which, in the non-degenerate case ν1 6= ν2 , gives
(see equations (2.7), (2.8), (2.9), and (2.10))
00
0
V (ν1 ,ν2 ) (x) = V (x) − 2 log W (ν1 ,ν2 ) (x) = V (x) − 2 (Eν2 − Eν1 )/(wν2 (x) − wν1 (x)) ,
ψν(ν21 ) (x) = (wν1 (x) − wν2 (x))ψν2 (x) = W (ψν1 , ψν2 | x)/ψν1 (x).
Note that in the degenerate case ν1 = ν2 , we have (see equation (2.6))
ψν(ν11 ) (x) = 1/ψν1 (x),
wν(ν11 ) (x) = −wν1 (x),
(3.1)
Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
(ν ) and by applying the DBT A wν11
5
= A(−wν1 ) to V (ν1 ) , we recover simply the initial potential V ,
V (ν1 ,ν1 ) (x) = V (ν1 ) (x) − 2wν0 1 (x) = V (x).
The confluent case corresponds to the limit ν2 → ν1 . As proven by Fernández et al. [7, 20],
the confluent extended potential and its eigenstates admit the following integral representations
00
Z x
(ν1 ,ν1 )
2
e
(3.2)
(x) = V (x) − 2 log
dtψν1 (t) − W0
V
x0
and
ψν21 (x)
(ν )
ψk 1 (x).
2
x0 dtψν1 (t) − W0
(ν ,ν )
ψek 1 1 (x) = (Eν1 − Ek )ψk (x) − R x
(3.3)
Both depend on an arbitrary real parameter W0 and for an adapted range of W0 values, the
extended potential is regular. In fact, the formula for the potential (3.2) already appears in
many previous works, for instance in a 1986 paper of Luban and Pursey [33] and a few years
later in [31]. The Matveev formulas [27, 34, 35] for the two-step case
!#00
"
∂ψ
(x)
ν
(ν
,ν
)
Ve 1 1 (x) = V (x) − 2 log W ψν1 ,
|x
,
∂Eν
ν=ν1
!
!
.
∂ψ
(x)
∂ψ
(x)
(ν
,ν
)
ν
ν
1
1
, ψk | x
W ψν1 ,
|x ,
(3.4)
ψek
(x) = W ψν1 ,
∂Eν
∂Eν
ν=ν1
ν=ν1
which express the confluent extension and its eigenstates in terms of generalized Wronskians
(which are in fact two-way, or double Wronskians [51]) can be viewed as associated to a particular
choice of the W0 constant. Indeed, if we consider the indexed family of RS functions wν (x) as
satisfying a prescribed initial condition in x0 , we have, in the confluent limit, wν2 (x) → wν1 (x).
It results from equation (3.1) that
!0
. ∂w (x) ν
.
V (ν1 ,ν1 ) (x) = V (x) − 2 1
∂Eν
ν=ν1
But we can readily verify that
"
∂wν (x)
1/
= log W
∂Eν
ν=ν1
ψν1 ,
∂ψν (x)
∂Eν
!#0
|x
ν=ν1
and since in this case we also have [36]
Z x
1
∂wν (x)
= 2
dtψν21 (t),
∂Eν
ψ
(x)
x0
ν1
ν=ν1
we see that equation (3.4) corresponds to equations (3.2) and (3.3) with W0 = 0.
It has to be noticed that the degenerate extension V (ν1 ,ν1 ) (x) can be recovered from the
confluent one Ve (ν1 ,ν1 ) (x) by taking the singular limit value W0 → ∞.
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 [7, 20].
The preceding results can be in fact integrated within the standard DBT scheme simply
using the DBT in its completed form (see equation (2.6)) as in [13, 31]. Indeed, by applying the
DBT A(wν ), we generate first the one-step (possibly singular) extension
V (ν) (x) = V (x) + 2wν0 (x) = V (x) − 2(log ψν (x))00 .
6
Y. Grandati and C. Quesne
(ν)
Since ψν = 1/ψν is an eigenfunction of V (ν) for the eigenvalue Eν , the most general eigenfunction (up to a multiplicative factor) of V (ν) for the same eigenvalue is
!
Rx
Z x
λ1 + x0 dtψν2 (t)
1
(ν)
(ν)
Ψν (x; λ1 ) = ψν (x) λ1 +
dt (ν)
,
λ1 ∈ R,
(3.5)
=
ψν (x)
x0
(ψν (t))2
and the corresponding RS function is
!#0
"
Rx
2 (t)
λ
+
dtψ
1
ψν2 (x)
ν
x0
Rx
.
Wν(ν) (x; λ1 ) = − log
= −wν (x) −
ψν (x)
λ1 + x0 dtψν2 (t)
We can now use this general solution as seed function for the second DBT. Then ap(ν) plying A Wν
to V (ν) , we obtain the following second extension
00
00
Ve (ν,ν) (x; λ1 ) = V (ν) (x) − 2 log Ψ(ν)
= V (x) − 2 log ψν (x)Ψ(ν)
ν (x; λ1 )
ν (x; λ1 )
!0
00
Z x
ψν2 (x)
2
Rx
= V (x) − 2 log λ1 +
dtψν (t)
= V (x) − 2
, (3.6)
λ1 + x0 dtψν2 (t)
x0
and we recover the first Fernández formula (3.2) with λ1 = −W0 . As for the eigenfunctions
of Ve (ν,ν) , they are given by (µ 6= ν)
b Wν(ν) ψµ(ν) (x) = (Eν − Eµ )ψµ (x) − W (ψν , ψµ | x) .
ψeµ(ν,ν) (x; λ1 ) = A
(ν)
Ψν (x; λ1 )
3.2
(3.7)
General multi-step conf luent chains
Fernández and Salinas-Hernández [21] have also considered the so-called “hyperconfluent” case
corresponding to a three-step confluent DBT, for which they have extended the previous formulas, equations (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).
| {z }
l times
(ν)
A(Wν )
(ν)
Ψν
In the three-step case, the image of
(see equation (3.5)) by the DBT
being
2)
(ν)
(ν
= 1/Ψν , the general eigenfunction of V
(x; λ1 ) (see equation (3.6)) associated to the
eigenvalue Eν is
2
Rx
(ν)
λ
+
dt
Ψ
(t;
λ
)
ν
2
1
2
x
)
0
,
λ1 , λ2 ∈ R,
Ψ(ν
ν (x; Λ2 ) =
(ν)
Ψν (x; λ1 )
(ν 2 )
ψν
where we have used the notation Λm = (λ1 , . . . , λm ). The next extension generated by the DBT
(ν 2 ) A Wν
is then
00
3
(ν 2 )
Ve (ν ) (x; Λ2 ) = V (x) − 2 log ψν (x)Ψ(ν)
ν (x; λ1 )Ψν (x; Λ2 )
Z x
2 00
00
(ν)
= V (x) − 2(log ψν (x)) − 2 λ2 +
dt Ψν (t; λ1 )
.
x0
We recover the “hyperconfluent” third-order superpartner of V (x) as obtained by Fernández
and Salinas-Hernández [21]. Within this scheme, the generalization is immediate and repeating the procedure m times, we obtain for the hyperconfluent mth -order extension of V (x) the
Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
7
expression

Ve
(ν m )
(x; Λm−1 ) = V (x) − 2 log
m−1
Y
00
(ν j )
Ψν
(x; Λj ) = V (x) − 2
m−1
X
j=0
00
j)
log Ψ(ν
ν (x; Λj ) ,
j=0
with the following recurrence relation for the successive seed functions
Z x
2
(ν k−1 )
dt Ψν
(t; Λk−1 )
λk +
k)
x0
,
Ψ(ν
(x; Λk ) =
ν
(ν k−1 )
Ψν
(x; Λk−1 )
(0)
where Ψν (x; Λ0 ) = ψν (x).
In other words, whenever m is even (m = 2k)
"k−1 Z
Y
(x; Λ2k−1 ) = V (x) − 2 log
λ2l+1 +
(
Ve
(ν 2k )
x
(ν 2l )
dt Ψν
2
(t; Λ2l )
#)00
x0
l=0
and whenever m is odd (m = 2k + 1)
(
Ve
(ν 2k+1 )
"
(x; Λ2k ) = V (x) − 2 log ψν (x)
k Y
l=1
Z
x
λ2l +
(ν 2l−1 )
dt Ψν
(t; Λ2l−1 )
2
#)00
m
b Wν(ν
The eigenstates of Ve (ν ) can be obtained by successive applications of the A
(the product being ordered in decreasing order),
(ν
ψek
m)
(x; Λm−1 ) =
m
Y
.
x0
l)
operators
b W (ν l−1 ) ψk (x).
A
ν
l=1
A direct application of the Crum Wronskian formulas [14] to these general, parameterdependent, confluent extensions is obviously not possible and, as mentioned above, the Matveev
formulas [27, 34, 35] correspond only to a particular choice of the λj parameters. Nevertheless,
these extended potentials are amenable to other (standard) Wronskian formulas [7, 20, 48].
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.
4
4.1
Two-step conf luent rational extensions of the trigonometric
Darboux–Pöschl–Teller (TDPT) potential
General scheme
The trigonometric Darboux–Pöschl–Teller (TDPT) potential (with zero ground-state energy) is
defined on x ∈ ]0, π/2[ by
V (x; α, β) =
with α, β > 1/2.
(α + 1/2)(α − 1/2) (β + 1/2)(β − 1/2)
+
− (α + β + 1)2 ,
cos2 x
sin2 x
8
Y. Grandati and C. Quesne
Its physical spectrum, associated to the asymptotic Dirichlet boundary conditions
π −
+
ψ(0 ; α, β) = 0 = ψ
; α, β ,
2
is given in terms of Jacobi polynomials [18, 29, 50]
Pn(α,β) (z) =
n
(−1)n Γ(n + β + 1) X
n Γ(n + α + β + 1 + k)
(−1)k
(1 + z)k ,
n!Γ(n + α + β + 1)
k
2k Γ(β + 1 + k)
k=0
by
En (α, β) = (αn + βn + 1)2 − (α + β + 1)2 = 4n(α + β + 1 + n),
ψn (x; α, β) = ψ0 (x; α, β)Pn(α,β) (z),
n ∈ N,
with z = cos 2x ∈ ]−1, 1[, (αn , βn ) = (α + n, β + n), and
ψ0 (x; α, β) = (1 − z)(α+1/2)/2 (1 + z)(β+1/2)/2 .
In the following, in order to get rational extensions, we consider the case where α and β are
integers: α = N ≥ 1, β = M ≥ 1.
If we choose as initial seed function an eigenstate ψn (x; N, M ) of V (x; N, M ), by taking
x0 = π/2 (z0 = −1), the quantity
Z x
Z
2
1 z
Qn(N,M ) (z) =
dξψn2 (ξ; N, M ) = −
dζ(1 − ζ)N (1 + ζ)M Pn(N,M ) (ζ)
2 −1
π/2
is a polynomial of degree N + M + 2n + 1 in z with [18, 50]
Z
2
1 1
(N,M )
Qn
(1) = −
dζ(1 − ζ)N (1 + ζ)M Pn(N,M ) (ζ)
2 −1
=−
2N +M
(n + N )!(n + M )!
.
2n + N + M + 1 n!(n + N + M )!
(4.1)
Note the following recurrence
(N +1,M +1)
Qn−1
(1) =
4n
Q(N,M ) (1).
n+N +M +1 n
(4.2)
2
From equation (3.6), we then obtain for the confluent two-step extension Ve (n ) ,
00
2
)
Ve (n ) (x; N, M, λ1 ) = V (x; N, M ) − 2 log λ1 + Q(N,M
(z)
n
!
N +1/2 (1 + z)M +1/2 P (N,M ) (z) 2
(1
−
z)
d
n
1/2
,
= V (x; N, M ) + 4 1 − z 2
(N,M )
dz
λ1 + Qn
(z)
(N,M )
2
where Ve (n ) (x; N, M, λ1 ) constitutes a rational extension of V (x) (in the z variable). Qn
(z)
is strictly decreasing on the interval ]−1, 1[ and keeps a negative value, varying from 0 to
(N,M )
Qn
(1) < 0 when z runs through ]−1, 1[. Consequently, when
)
λ1 ∈ ] − ∞, 0] ∪ −Q(N,M
(1), +∞ ,
(4.3)
n
(N,M )
then λ1 + Qn
(z) keeps a constant sign, strictly negative or strictly positive respectively, and
2
Ve (n ) (x; N, M, λ1 ) is regular.
Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
9
In this case, its eigenfunctions for k 6= n are given by (see equations (3.5) and (3.7))
W (ψn , ψk | x)
2
(n )
ψek (x; N, M, λ1 ) = [En (N, M ) − Ek (N, M )]ψk (x) −
(n)
,
Ψn (x; N, M, λ1 )
with
(N,M )
−(N +1/2)/2
Ψ(n)
(1 + z)−(M +1/2)/2
n (x; N, M, λ1 ) = (1 − z)
λ1 + Qn
(z)
(N,M )
Pn
(z)
and
(N,M )
W (ψn , ψk | x) = −(1 − z)N +1 (1 + z)M +1 Pn,k
(z),
(α,β)
where (P−1 (z) = 0)
(N,M )
Pn,k
(N +1,M +1)
(z) = (k + N + M + 1)Pn(N,M ) (z)Pk−1
(N +1,M +1)
− (n + N + M + 1)Pn−1
(z)
(N,M )
(z)Pk
(z)
is an exceptional Jacobi polynomial in the broad sense of the term.
Hence, for k 6= n,
2
(n )
ψek (x; N, M, λ1 ) = [En (N, M ) − Ek (N, M )]ψk (x; N, M )
(N,M )
+ (1 − z)
(3N +5/2)/2
(1 + z)
(3M +5/2)/2
Pn,k
λ1 +
(N,M )
(z)Pn
(z)
(N,M )
Qn
(z)
,
that is
2
(n2 )
ψek (x; N, M, λ1 )
= (1 − z)(N +1/2)/2 (1 + z)(M +1/2)/2
(n )
PeN,M,k (z; λ1 )
(N,M )
λ1 + Qn
,
(z)
where
(n2 )
(N,M )
)
PeN,M,k (z; λ1 ) = 4(n − k)(n + k + N + M + 1)Pk
(z) λ1 + Q(N,M
(z)
n
(N,M )
+ (1 − z)N +1 (1 + z)M +1 Pn,k
(z)Pn(N,M ) (z).
Moreover
2
ψen(n ) (x; N, M, λ1 ) = 1/Ψ(n)
n (x; N, M, λ1 )
(N,M )
= (1 − z)
(N +1/2)/2
(1 + z)
(M +1/2)/2
Pn
λ1 +
(z)
(N,M )
Qn
(z)
,
2
which is a normalizable eigenstate of Ve (n ) (x; N, M, λ1 ). This corresponds to defining
2
(n )
PeN,M,n (z; λ1 ) = Pn(N,M ) (z).
2
(n )
ψek (x; N, M, λ1 ) tends to zero at z = −1 and z = 1 (i.e., x = π/2 and x = 0) and is then
2
an admissible eigenstate of Ve (n ) (x; N, M, λ1 ) for every k ≥ 0. The potentials V (x; N, M ) and
2
Ve (n ) (x; N, M, λ1 ) are therefore strictly isospectral.
10
Y. Grandati and C. Quesne
2
(n )
The orthogonality conditions between eigenstates imply that the PeN,M,k (z; λ1 ) constitute
a family of orthogonal polynomials (indexed by k ∈ N) on ]−1, 1[ with respect to the measure
(n2 )
µN,M (z; λ1 ) =
1 (1 − z)N (1 + z)M
2 .
)
2 λ1 + Q(N,M
(z)
n
2
It is worth observing here that the confluent two-step extension Ve (n ) (x; N, M, λ1 ) may be
considered as a special case of one of those with general parameters that have been built by
Contreras-Astorga and Fernández in [12, Section 3.2.3(b)] (namely the third one given in equation (3.43)), whenever their parameters λ, ν assume the half-integer values λ = N + 1/2 and
ν = M + 1/2.
4.2
Examples n = 0
We consider the n = 0 case. Then
(N,M )
Q0
(z)
1
=−
2
Z
z
N
M
dζ(1 − ζ) (1 + ζ)
= −(z + 1)
−1
M +1
N
X
2N −k−1 (−1)k N
(z + 1)k
M +k+1
k
k=0
and
2N +1
2M +1
(1 − z)
(1 + z)
2
Ve (0 ) (x; N, M, λ1 ) = V (x; N, M ) + 2
2
(N,M )
λ1 + Q0
(z)
M − N − (N + M + 1)z
+ 4(1 − z)N (1 + z)M
.
(N,M )
λ 1 + Q0
(z)
Moreover
2
(02 )
ψek (x; N, M, λ1 )
= (1 − z)
(N +1/2)/2
(1 + z)
(M +1/2)/2
(0 )
PeN,M,k (z; λ1 )
(N,M )
λ1 + Q0
where
(02 )
(N,M )
(N,M )
PeN,M,k (z; λ1 ) = −4k(k + N + M + 1)Pk
(z) λ1 + Q0
(z)
(N,M )
+ (1 − z)N +1 (1 + z)M +1 P0,k
(z),
k 6= 0,
(02 )
PeN,M,0 (z; λ1 ) = 1,
(N,M )
with (P−1
(N,M )
P0,k
4.2.1
(z) = 0)
(N +1,M +1)
(z) = (k + N + M + 1)Pk−1
(z).
The N = M = 1 case
The results read
Q0
1
1
(z) = − (z + 1)2 1 − (z + 1) ,
2
3
e (02 )
z 1 − z2
(x; 1, 1, λ1 ) = V (x; 1, 1) − 12
λ1 − 12 (z + 1)2 + 16 (z + 1)3
(1,1)
V
(z)
,
Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
+ 2
1 − z2
3
λ1 − 21 (z + 1)2 + 16 (z + 1)3
2
or
2
Ve (0 ) (x; 1, 1, λ1 ) =
3
3
sin2 2x cos 2x
+
−
9
−
12
2
4 sin x 4 cos2 x
λ1 − 2 cos4 x + 43 cos6 x
+2
sin6 2x
λ1 − 2 cos4 x + 43 cos6 x
2
with λ1 ∈ ]−∞, 0] ∪ ] 23 , +∞[, and
(02 )
ψek
(x; 1, 1, λ1 ) = (1 − z)3/4 (1 + z)3/4
(02 )
Pe1,1,k (z; λ1 )
λ1 − 12 (z + 1)2 + 16 (z + 1)3
,
where
(02 )
Pe1,1,k (z; λ1 )
= −4k(k +
(1,1)
3)Pk (z)
1
1
λ1 − (z + 1)2 + (z + 1)3
2
6
(2,2)
+ (k + 3)(1 − z)2 (1 + z)2 Pk−1 (z),
k 6= 0,
(02 )
Pe1,1,0 (z; λ1 ) = 1.
4.2.2
The N = 2, M = 1 case
The results read
2
1
2
= −(z + 1) 1 − (z + 1) + (z + 1) ,
3
8
(1 + 4z)(1 − z)2 (1 + z)
2
Ve (0 ) (x; 2, 1, λ1 ) = V (x; 2, 1) − 4
λ1 − (z + 1)2 + 23 (z + 1)3 − 18 (z + 1)4
(2,1)
Q0 (z)
2
+ 2
(1 − z)5 (1 + z)3
λ1 − (z + 1)2 + 23 (z + 1)3 − 18 (z + 1)4
2
or
2
Ve (0 ) (x; 2, 1, λ1 ) =
15
3
(1 − 4 cos 2x) sin4 x cos2 x
−
16
−
32
+
6
8
4 sin2 x 4 cos2 x
λ1 − 4 cos4 x + 16
3 cos x − 2 cos x
+ 512
sin10 x cos6 x
λ1 − 4 cos4 x +
16
3
cos6 x − 2 cos8 x
2
with λ1 ∈ ]−∞, 0] ∪ ] 32 , +∞[, and
2
(02 )
ψek (x; 2, 1, λ1 )
= (1 − z)
5/4
(0 )
Pe2,1,k (z; λ1 )
3/4
(1 + z)
λ1 − (z + 1)2 + 23 (z + 1)3 − 18 (z + 1)4
,
where
2
1
(02 )
(2,1)
2
3
4
e
P2,1,k (z; λ1 ) = −4k(k + 4)Pk (z) λ1 − (z + 1) + (z + 1) − (z + 1)
3
8
(3,2)
+ (k + 4)(1 − z)3 (1 + z)2 Pk−1 (z),
(02 )
Pe2,1,0 (z; λ1 )
= 1.
k 6= 0,
11
12
Y. Grandati and C. Quesne
4.2.3
The N = 1, M = 2 case
The results read
1 1
= −(z + 1)
− (z + 1) ,
3 8
(1 − 4z)(1 − z)(1 + z)2
2
Ve (0 ) (x; 1, 2, λ1 ) = V (x; 1, 2) + 4
λ1 − 13 (z + 1)3 + 18 (z + 1)4
(1,2)
Q0 (z)
3
+ 2
(1 − z)3 (1 + z)5
λ1 − 31 (z + 1)3 + 18 (z + 1)4
2
or
2
Ve (0 ) (x; 1, 2, λ1 ) =
15
(1 − 4 cos 2x) sin2 x cos4 x
3
+
−
16
+
32
4 sin2 x 4 cos2 x
λ1 − 83 cos6 x + 2 cos8 x
+ 512
sin6 x cos10 x
λ1 − 38 cos6 x + 2 cos8 x
2
with λ1 ∈ ]−∞, 0] ∪ ] 23 , +∞[, and
2
(02 )
ψek (x; 1, 2, λ1 )
= (1 − z)
3/4
(0 )
Pe1,2,k (z; λ1 )
5/4
(1 + z)
λ1 − 13 (z + 1)3 + 18 (z + 1)4
,
where
1
1
(02 )
(1,2)
Pe1,2,k (z; λ1 ) = −4k(k + 4)Pk (z) λ1 − (z + 1)3 + (z + 1)4
3
8
(2,3)
+ (k + 4)(1 − z)2 (1 + z)3 Pk−1 (z),
k 6= 0,
(02 )
Pe1,2,0 (z; λ1 ) = 1.
4.3
Shape invariance of the two-step conf luent rational extensions
of the TDPT potential
V (x; N, M ) is a translationally shape invariant potential with a SUSY partner
VSUSY (x; N, M ) = V (0) (x; N, M ) = V (x; N + 1, M + 1) + E1 (N, M )
and (the coefficient is readily established by a direct calculation)
W (ψ0 , ψn | x)
En (N, M )
ψn(0) (x; N, M ) =
= −
ψn−1 (x; N + 1, M + 1).
ψ0 (x; N, M )
4n
2
Since Ve (n ) (x; N, M, λ1 ) and V (x; N, M ) are strictly isospectral, it is natural to wonder
2
whether Ve (n ) (x; N, M, λ1 ) shares the same invariance property as V (x; N, M ).
2
Consider the SUSY partner of Ve (n ) (x; N, M, λ1 ). It is given by
00
2
(n2 )
(n2 )
VeSUSY (x; N, M, λ1 ) = Ve (n ) (x; N, M, λ1 ) − 2 log ψe0 (x; N, M, λ1 )
2
e(n ) (x; N, M, λ1 ) 00 ,
= V (x; N, M ) − 2 log ψn (x; N, M )Ψ(n)
n (x; N, M, λ1 )ψ0
where, for n ≥ 1,
2
(n )
ψe0 (x; N, M, λ1 ) = En (N, M )ψ0 (x; N, M ) −
W (ψn , ψ0 | x)
(n)
Ψn (x; N, M, λ1 )
Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
13
and, for n = 0,
1
2
(0 )
ψe0 (x; N, M, λ1 ) =
(0)
Ψ0 (x; N, M, λ1 )
.
Consequently, in the n = 0 case, we have
2
(0 )
VeSUSY (x; N, M, λ1 ) = V (x; N, M )
00
(0)
(02 )
− 2 log ψ0 (x; N, M )Ψ0 (x; N, M, λ1 )ψe0 (x; N, M, λ1 )
= V (x; N, M ) − 2[log ψ0 (x; N, M )]00 ,
that is
2
(0 )
VeSUSY (x; N, M, λ1 ) = V (0) (x; N, M ) = V (x; N + 1, M + 1) + E1 (N, M ).
Furthermore, in the n ≥ 1 case, we get
2
(n )
VeSUSY (x; N, M, λ1 ) = V (x; N, M ) − 2[log ψ0 (x; N, M )]00
00
W (ψ0 , ψn | x)
(n)
− 2 log En (N, M )Ψn (x; N, M, λ1 ) +
ψn (x; N, M )
ψ0 (x; N, M )
= V (x; N + 1, M + 1) + E1 (N, M )
En (N, M )
(n)
− 2 log En (N, M )Ψn (x; N, M, λ1 ) −
ψn−1 (x; N + 1, M + 1)
4n
00
+ log [ψn (x; N, M )] .
More precisely
2
(n )
VeSUSY (x; N, M, λ1 ) = V (x; N + 1, M + 1) − 2[log ψn−1 (x; N + 1, M + 1)]00 + E1 (N, M )
!00
(N,M )
λ 1 + Qn
(z) − ψn (x; N, M )ψn−1 (x; N + 1, M + 1)/4n
− 2 log
ψn−1 (x; N + 1, M + 1)
= V (n−1) (x; N + 1, M + 1) + E1 (N, M )
(N,M )
− 2 log
λ 1 + Qn
(z) − ψn (x; N, M )ψn−1 (x; N + 1, M + 1)/4n
ψn−1 (x; N + 1, M + 1)
!00
.
If, for an appropriate constant C, the following condition
)
λ1 + Q(N,M
(z) −
n
ψn (x; N, M )ψn−1 (x; N + 1, M + 1)
(N +1,M +1)
= C λ01 + Qn−1
(z)
4n
(4.4)
is satisfied, then
(N,M )
λ1 + Qn
(z) − ψn (x; N, M )ψn−1 (x; N + 1, M + 1)/4n
(n−1)
= CΨn−1 (x; N + 1, M + 1, λ01 )
ψn−1 (x; N + 1, M + 1)
and we obtain an enlarged shape invariance property
2
2
(n )
VeSUSY (x; N, M, λ1 ) = Ve ((n−1) ) (x; N + 1, M + 1, λ01 ) + E1 (N, M ).
(4.5)
The preceding condition (4.4) can be rewritten as
A(z) − B(z) = Cλ01 − λ1 ,
∀ z ∈ ]−1, 1[
(4.6)
14
Y. Grandati and C. Quesne
with
(N +1,M +1)
)
A(z) = Q(N,M
(z) − CQn−1
(z)
n
Z z h
2
1
(N +1,M +1)
=
(ζ)
dζ C(1 − ζ)N +1 (1 + ζ)M +1 Pn−1
2 −1
2 i
− (1 − ζ)N (1 + ζ)M Pn(N,M ) (ζ)
and
B(z) = ψn (x; N, M )ψn−1 (x; N + 1, M + 1)/4n
(1 − z)N +1 (1 + z)M +1 (N,M )
(N +1,M +1)
Pn
(z)Pn−1
(z).
4n
=
Equation (4.6) is equivalent to
d
d
A(z) =
B(z),
dz
dz
d
dz A(z)
where
(4.7)
is given by
(N +1,M +1) 2
2
d
2
2
(N,M )
A(z)
=
C
1
−
z
P
(z)
−
P
(z)
.
n
n−1
(1 − z)N (1 + z)M dz
As for
d
dz B(z),
using the derivation formula [18, 50]
d (N,M )
N + M + n + 1 (N +1,M +1)
Pn
(z) =
Pn−1
(z),
dz
2
(4.8)
it can be expressed as
4n
d
d (N +1,M +1)
B(z) = 1 − z 2 Pn(N,M ) (z) Pn−1
(z)
(1 −
+
dz
dz
N + M + n + 1 (N +1,M +1) 2
+ 1 − z2
Pn−1
(z)
2
(N +1,M +1)
+ [(M − N ) − z(N + M + 2)]Pn(N,M ) (z)Pn−1
(z),
z)N (1
z)M
that is
N + M + n + 1 (N +1,M +1) 2
d
B(z) = 1 − z 2
Pn−1
(z)
(1 −
+
dz
2
d (N +1,M +1)
+ 1 − z 2 Pn(N,M ) (z) Pn−1
(z)
dz
(N +1,M +1)
+ [(M − N ) − z(N + M + 2)]Pn(N,M ) (z)Pn−1
(z).
4n
z)N (1
z)M
But the differential equation satisfied by the Jacobi polynomials is [18, 50]
d2 (N,M )
d
P
(z) + [(M − N ) − z(N + M + 2)] Pn(N,M ) (z)
dz 2 n
dz
= −n(N + M + n + 1)Pn(N,M ) (z),
1 − z2
which, combined with equation (4.8), gives
1 − z2
d (N +1,M +1)
(N +1,M +1)
P
(z) + [(M − N ) − z(N + M + 2)]Pn−1
(z) = −2nPn(N,M ) (z).
dz n−1
Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
15
Consequently
4n
d
B(z)
(1 −
+
dz
2
N + M + n + 1 (N +1,M +1) 2
Pn−1
(z) ,
= −2n Pn(N,M ) (z) + 1 − z 2
2
z)N (1
z)M
or
2
d
B(z)
(1 −
+
dz
N + M + n + 1 (N +1,M +1) 2
2
= 1 − z2
Pn−1
(z) − Pn(N,M ) (z) .
4n
To satisfy the equality (4.7), we then must choose
z)N (1
z)M
N +M +n+1
.
4n
In this case, equation (4.6) simply becomes
C=
N +M +n+1 0
λ1 − λ1 .
4n
In the limit z → 1− , we obviously get
A(z) − B(z) =
B(1) = 0,
and (see equations (4.1) and (4.2))
)
A(1) = Q(N,M
(1) −
n
N + M + n + 1 (N +1,M +1)
Qn−1
(1) = 0,
4n
which implies
4n
λ1 .
(4.9)
N +M +n+1
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 λ01 given in equation (4.9). Note that
2
the latter relation ensures that the domain of λ1 values for which Ve (n ) (x; N, M, λ1 ) is regular
2
corresponds exactly to the domain of λ01 values for which Ve ((n−1) ) (x; N + 1, M + 1, λ01 ) is also
regular (see equations (4.2) and (4.3)).
In the n = N = M = 1 case, such a property can be directly verified as follows. We get
λ01 =
(2,2)
P0
(1,1)
(z) = 1,
P1
(z) = 2z,
C = 1, λ01 = λ1 , as well as
Z
2
1 z
(1,1)
(2,2)
A(z) = Q1 (z) − Q0 (z) =
dζ 1 − ζ 2 − 4ζ 2 1 − ζ 2
2 −1
Z z
1
2
1
=
dζ 1 − 6ζ 2 + 5ζ 4 = z 1 − z 2
2 −1
2
and
1 − z2
B(z) =
4
Hence the identity
A(z) = B(z),
2
2z.
∀ z ∈ ]−1, 1[,
is satisfied, which implies that
2
2
(1 )
VeSUSY (x; 1, 1, λ1 ) = Ve ((0) ) (x; 2, 2, λ1 ) + E1 (1, 1).
16
Y. Grandati and C. Quesne
5
Two-step conf luent rational extensions
of the isotonic potential
5.1
General scheme
The isotonic oscillator potential (with zero ground-state energy E0 = 0) is defined on the positive
half-line ]0, +∞[ by
V (x; ω, α) =
ω 2 2 (α + 1/2)(α − 1/2)
x +
− ω(α + 1),
4
x2
|α| > 1/2.
If we add Dirichlet boundary conditions at zero and infinity and if we suppose α > 1/2, it
has the following spectrum (z = ωx2 /2)
ψn (x; ω, α) = z (α+1/2)/2 e−z/2 Lαn (z),
En (ω) = 2nω,
n ≥ 0,
where
Lαn (z) =
n
X
(−1)k (n + α) · · · (k + 1 + α)
k=0
(n − k)!
k!
zk
(5.1)
is the usual Laguerre polynomial [18, 50].
In the following, in order to get rational extensions, we consider the case where α is an integer:
α = N ≥ 1.
If we choose as initial seed function an (unnormalized) eigenstate ψn (x; ω, N ) of V (x; ω, N ),
by taking x0 = 0 we get
Z +∞
Z x
2
2
dtψn (t; ω, N ) = kψn k −
dtψn2 (t; ω, N )
x
0
Z +∞
2
1
= kψn k2 − √
dζζ N LN
n (ζ) exp(−ζ).
2ω z
But
FnN (z)
1
= −√
2ω
Z
+∞
dζζ N LN
n (ζ)
z
2
1
exp(−ζ) = √ exp(−z)QN
n (z),
2ω
(5.2)
where
QN
n (z) = −
NX
+2n
j=0
dj N N 2 z Ln (z)
dz j
(5.3)
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 λ1 = −kψn k2 . Equation (3.6)
then becomes
00
2
Ve (n ) x; ω, N, −kψn k2 = V (x; ω, N ) − 2 log FnN (z)
!
N +1/2 LN (z) 2
z
d
n
= V (x; ω, N ) − 4ωz 1/2
,
dz
QN
n (z)
2
where Ve (n ) (x; ω, N, −kψn k2 ) constitutes a rational extension of V (x; ω, N ). Since FnN (z) is a
negative, strictly increasing function of z, such that
lim FnN (z) = 0,
x→+∞
Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
17
N
QN
n (z) keeps a constant strictly negative sign, i.e., Qn (z) has no zero on the positive half-line.
2)
(n
2
e
This shows that V
(x; ω, N, −kψn k ) is also regular.
Let us note too that [18, 50]
Z +∞
2
1
1 (n + N )!
FnN (0) = − √
dζζ N LN
< 0,
(5.4)
n (ζ) exp(−ζ) = − √
n!
2ω 0
2ω
or
QN
n (0) = −
(n + N )!
.
n!
(5.5)
2
The eigenfunctions of Ve (n ) (x; ω, N, −kψn k2 ) are given by (see equation (3.7))
2)
(n
ψek
x; ω, N, −kψn k2 = [En (ω) − Ek (ω)]ψk (x; ω, N ) −
W (ψn , ψk | x)
,
(n)
Ψn (x; ω, N, −kψn k2 )
k 6= n,
where
Rx
Ψ(n)
n
2
dtψn2 (t; ω, N ) − kψn k2
ψn (x; ω, N )
N
Fn (z)
1
QN
n (z)
=
= √ z −(N +1/2)/2 exp(−z/2) N
ψn (x; ω, N )
Ln (z)
2ω
x; ω, N, −kψn k ) =
0
(5.6)
and
W (ψn , ψk | x) =
√
2ωz N +1 exp(−z)LN
n,k (z),
with
N +1
N +1
N
N
LN
n,k (z) = Ln−1 (z)Lk (z) − Ln (z)Lk−1 (z).
Consequently
2)
(n
ψek
x; ω, N, −kψn k2 = [En (ω) − Ek (ω)]ψk (x; ω, N )
− 2ωz (3N +5/2)/2 exp(−z/2)
N
LN
n,k (z)Ln (z)
QN
n (z)
,
that is,
2)
(n
ψek
2ωz (N +1/2)/2 exp(−z/2) (n2 )
e
x; ω, N, −kψn k2 =
L
k,N (z),
QN
(z)
n
where
2
N +1 N
e (n ) (z) = (n − k)LN (z)QN
L
Ln,k (z)LN
n (z) − z
n (z).
k
k,N
2
(n )
ψek (x; ω, N, −kψn k2 ) decreases exponentially to zero at infinity and also tends to zero at
2
the origin (see equation (5.5)). It is therefore an admissible eigenstate of Ve (n ) (x; ω, N, −kψn k2 )
for every k 6= n.
Note that in this case
LN (z)
2
ψen(n ) x; ω, N, −kψn k2 = 1/Ψn(n) x; ω, N, −kψn k2 ∝ z (N +1/2)/2 exp(z/2) nN
Qn (z)
(5.7)
18
Y. Grandati and C. Quesne
2
is not an eigenstate since it is not normalizable. This implies that Ve (n ) (x; ω, N, −kψn k2 ) and
(ν)
V (x; ω, N ) are only quasi-isospectral, the two-step confluent DBT A(Wν ) ◦ A(wν ) being now
state-deleting.
2
e (n ) (z) with k 6= n,
The orthogonality conditions between eigenstates imply that the set of L
k,N
constitute a family of orthogonal polynomials, indexed by k ∈ N, on the positive half-line with
respect to the mesure
(n2 )
µN (z) =
5.2
z N exp(−z)
2 .
QN
(z)
n
Examples
5.2.1
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 one(0)
step DBT A(w0 ) and A(W0 ). The first one is the usual SUSY partnership and, due to the
shape invariance property of the isotonic potential, we have
V (0) (x; ω, N ) = V (x; ω, N + 1) + 2ω,
where V (0) (x; ω, N ) admits the following eigenstates
(0)
ψk (x; ω, N ) ∼ ψk−1 (x; w, N + 1),
for the respective energies Ek (ω), k ≥ 1.
(0) (0)
The second DBT A W0
is associated to the seed function Ψ0 (x; ω, N, −kψ0 k2 ), which
is a formal eigenfunction of V (0) (x; ω, N ) for the eigenvalue E0 = 0 and which is then in the
disconjugacy sector of V (0) (x; ω, N ). But
!
Z x
1
(0)
(0)
dt (0)
Ψ0 (x; ω, N, λ1 ) = ψ0 (x; ω, N ) λ1 +
2 ,
x0
ψ0 (t; ω, N )
where
(0)
ψ0 (x; ω, N ) ∼ 1/ψ0 (x; ω, N ) = z −(N +1/2)/2 e+z/2 .
By taking x0 = 0 and λ1 = − kψ0 k2 , we obtain
(0)
Ψ0
1
x; ω, N, −kψ0 k2 = √ z −(N +1/2)/2 exp(−z/2)QN
0 (z),
2ω
(5.8)
where
QN
0 (z)
N
N
X
X
zl
dj N =−
z
=
−N
!
,
dz j
l!
j=0
l=0
which gives
(0)
Ψ0
N
X
N!
zl
x; ω, N, −kψ0 k2 = − √ exp(−z/2)z −(N +1/2)/2
.
l!
2ω
l=0
(5.9)
Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
19
(0)
Equation (5.9) shows that Ψ0 (x; ω, N, −kψ0 k2 ) 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 E0 = 0) [6, 30].
(0)
Consequently Ψ0 (x; ω, N, −kψ0 k2 ) coincides necessarily (up to a constant multiplicative
factor) with a type II seed function [22] of V (x; ω, N + 1) + 2ω, which is a formal eigenfunction
for the eigenvalue E−1 + 2ω, namely
(0)
Ψ0
−1
x; ω, N, −kψ0 k2 ∼ φN,− (x; ω, N + 1) = z −(N +1/2)/2 e−z/2 L−N
(z),
N
where φn,− is defined by
φn,− (x; ω, α) = ψn (x; ω, −α).
Note that this result can be recovered directly by using the identity (see equation (5.1))
−1
L−N
(z) = (−1)N
N
N
X
zl
l=0
l!
,
in equation (5.9).
(0) Applying then the DBT A W0
= A(vN,− (x; ω, N + 1)) (vN,− = −φ0N,− /φN,− ) to V (0) (x;
ω, N ) = V (x; ω, N + 1) + 2ω, we arrive at a two-step confluent extended potential
2
Ve (0 ) x; ω, N, −kψ0 k2 = V (N,−) (x; ω, N + 1) + 2ω,
which coincides (up to a constant shift) with the usual extensions obtained with type II seed
functions [22].
Explicitly, we get
e (02 )
V
x; ω, N, −kψ0 k
2
!#
"
N
. X
4ω 1/2 d
l
N +1/2
= V (x; ω, N ) +
z
z /l!
z
N!
dz
l=0
and
(0
ψek
2)
(N +1/2)/2 exp(−z/2)
2
e (0 ) (z) z
x; ω, N, −kψ0 k2 ∼ L
,
k,N
N
P
l
z /l!
k ≥ 1.
l=0
In the N = 1 case, this gives
ω2 2
3
4ω
16ω
2
Ve (0 ) x; ω, 1, −kψ0 k2 =
x + 2+
−
= V (1,−) (x; ω, 2) + 2ω
2
4
4x
ωx + 2 (ωx2 + 2)2
and
(0
ψek
2)
2ωz 3/4 exp(−z/2) (02 )
e (z),
x; ω, 1, −kψ0 k2 ∼
L
k,1
z+1
with
2
e (0 ) (z) = kL1 (z)(z + 1) + z 2 L2 (z).
L
k
k−1
k,1
k ≥ 1,
20
Y. Grandati and C. Quesne
5.2.2
The n = 1 case
We have in this case
QN
1 (z)
= −z
N +2
+ Nz
N +1
− (N + 1)!
N
X
zj
j=0
j!
and
z N (N + 1 − z) 2
N + 21 (N + 1) − N + 52 z
Ve (1 ) x; ω, N, −kψ1 k2 = V (x; ω, N ) − 4ω
N
Q1 (z)


N
−1 j
X
z N +1 (N + 1 − z)2 
z
.
(N + 2)z N +1 − N (N + 1)z N + (N + 1)!
− 4ω
N 2
j!
Q1 (z)
j=0
2
The eigenstates of Ve (1 ) are (k 6= 1)
(1
ψek
2)
2
x; ω, N, −kψ1 k
= (1 − k)ψk (x; ω, N ) − z
(3N +5/2)/2
exp(−z/2)
N
LN
1,k (z)L1 (z)
QN
1 (z)
,
that is,
(1
ψek
2)
2ωz (N +1/2)/2 exp(−z/2) (12 )
e (z),
L
x; ω, N, −kψ1 k2 =
k,N
QN
1 (z)
where
2
e (1 ) (z) = z N +1 (N + 1 − z)2 LN +1 (z)
L
k−1
k,N

+ kz N +2 − (N k + 1)z N +1 + (k − 1)(N + 1)!
N
X
zj
j=0
j!

 LN
k (z).
In particular, for N = 1
Q11 (z) = −z 3 + z 2 − 2z − 2
and
2
Ve (1 ) x; ω, 1, −kψ1 k2 = V (x; ω, 1) + 6ω
z2 − 4
5z 2 − 4z − 2
−
20ω
2 + 2ω,
z 3 − z 2 + 2z + 2
z 3 − z 2 + 2z + 2
or
ω2 2
3
ω 2 x4 − 16
2
Ve (1 ) x; ω, 1, −kψ1 k2 =
x + 2 + 12ω 3 6
4
4x
ω x − 2ω 2 x4 + 8ωx2 + 16
2
4
5ω x − 8ωx2 − 8
− 320ω
2 .
ω 3 x6 − 2ω 2 x4 + 8ωx2 + 16
Moreover (k 6= 1)
(1
ψek
2)
2ωz 3/4 exp(−z/2) e (12 )
x; ω, 1, −kψ1 k2 = − 3
L (z),
z − z 2 + 2z + 2 k,1
where
2
e (1 ) (z) = kz 3 − (k + 1)z 2 + 2(k − 1)z + 2(k − 1) L1 (z) + z 2 (2 − z)2 L2 (z).
L
k
k−1
k,1
Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
21
The polynomial
2
e (1 ) (z) = − z 2 + 2z + 2
L
0,1
2
is associated to the ground state of Ve (1 ) x; ω, 1, −kψ1 k2 and
2
e (1 ) (z) = − 1 z 4 + 4z 2 − 12 ,
L
2,1
2
√
which has a single zero at 2 on ]0, +∞[, corresponds to the first excited eigenstate.
5.3
Shape invariance of the two-step conf luent rational extensions
of the isotonic potential
V (x; ω, N ) is a translationally shape invariant potential with a SUSY partner
VSUSY (x; ω, N ) = V (0) (x; ω, N ) = V (x; ω, N + 1) + E1 (ω)
and (the coefficient is readily established by a direct calculation)
√
W (ψ0 , ψn | x)
= − 2ωψn−1 (x; ω, N + 1).
(5.10)
ψ0 (x; ω, N )
2
Considering the SUSY partner of Ve (n ) x; ω, N, − kψn k2 , let us deal separately with the
n ≥ 1 and n = 0 cases. For n ≥ 1, this partner is given by
ψn(0) (x; ω, N ) =
00
2
(n2 )
(n2 )
VeSUSY x; ω, N, −kψn k2 = Ve (n ) x; ω, N, −kψn k2 − 2 log ψe0
x; ω, N, −kψn k2
(5.11)
2
00
(n )
= V (x; ω, N ) − 2 log ψn (x; ω, N )Ψn(n) (x; ω, N, −kψn k2 )ψe0 (x; ω, N, −kψn k2 ) ,
where
2)
(n
ψe0
x; ω, N, −kψn k2 = En (ω)ψ0 (x; ω, N ) −
W (ψn , ψ0 | x)
(n)
Ψn
x; ω, N, −kψn k2
.
(5.12)
Inserting (5.12) into (5.11), we get
(n2 )
VeSUSY x; ω, N, −kψn k2 = V (x; ω, N ) − 2(log ψ0 (x; ω, N ))00
00
W (ψ0 , ψn | x)
(n)
2
− 2 log En (ω)Ψn x; ω, N, −kψn k +
ψn (x; ω, N )
ψ0 (x; ω, N )
= V (x; ω, N + 1) + E1 (ω)
n h
io00
√
2
− 2 log 2nωΨ(n)
x;
ω,
N,
−kψ
k
−
2ωψ
(x;
ω,
N
+
1)
ψ
(x;
ω,
N
)
.
n−1
n
n
n
More precisely, on using (5.6),
(n2 )
VeSUSY x; ω, N, −kψn k2 = V (x; ω, N + 1) − 2(log ψn−1 (x; ω, N + 1))00 + E1 (ω)
00
ψn (x; ω, N )ψn−1 (x; ω, N + 1)
N
√
− 2 log Fn (z) −
/ψn−1 (x; ω, N + 1)
2ωn
= V (n−1) (x; ω, N + 1) + E1 (ω)
00
ψn (x; ω, N )ψn−1 (x; ω, N + 1)
√
− 2 log FnN (z) −
/ψn−1 (x; ω, N + 1)
.
2ωn
22
Y. Grandati and C. Quesne
If, for an appropriate constant C, the following condition
FnN (z) −
ψn (x; ω, N )ψn−1 (x; ω, N + 1)
N +1
√
= CFn−1
(z)
2ωn
(5.13)
is satisfied, then
ψn (x; ω, N )ψn−1 (x; ω, N + 1) .
N
√
Fn (z) −
ψn−1 (x; ω, N + 1)
2ωn
(n−1)
= CΨn−1 x; ω, N + 1, −kψn−1 k2
and we obtain an enlarged shape invariance property
2
(n2 )
Ve
x; ω, N, −kψn k2 = Ve ((n−1) ) x; ω, N + 1, −kψn−1 k2 + E1 (ω).
SUSY
(5.14)
The preceding condition (5.13) can be rewritten as
A(z) − B(z) = 0,
∀ z ∈ ]−1, 1[,
(5.15)
with
A(z) =
FnN (z)
−
N +1
(z)
CFn−1
1
=√
2ω
Z
z
+∞
2
2 +1
dζ Cζ N +1 LN
− ζ N LN
exp(−ζ)
n (ζ)
n−1 (ζ)
and
B(z) =
+1
z N +1 e−z LN
(z)LN
ψn (x; ω, N )ψn−1 (x; ω, N + 1)
n−1 (z)
√
√n
=
.
2ωn
2ωn
Equation (5.15) is equivalent to the set of conditions
d
d
A(z) =
B(z)
dz
dz
(5.16)
A(0) = B(0).
(5.17)
and
Here
√
As for
d
dz A(z)
2ω
is given by
d
A(z) =
dz
d
dz B(z),
2
2 N
+1
− Cz LN
LN
z exp(−z).
n (z)
n−1 (z)
using the derivation formula [18, 50]
d N
+1
L (z) = −LN
n−1 (z),
dz n
it can be expressed as
√
2
d
d N
d N
N
N
2ωn B(z) = −z exp(−z)Ln (z) z 2 Ln (z) + (N + 1 − z) Ln (z)
dz
dz
dz
2
+1
− z N +1 exp(−z) LN
n−1 (z) .
The differential equation satisfied by the Laguerre polynomials is [18, 50]
z
d
d2 N
Ln (z) + (N + 1 − z) LN
(z) = −nLN
n (z),
2
dz
dz n
(5.18)
Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
23
which, inserted in equation (5.18), yields
√
2ω
d
B(z) =
dz
2 1
2 N
+1
LN
z exp(−z).
− z LN
n (z)
n−1 (z)
n
Hence, to satisfy the first condition (5.16), we must choose
C=
1
.
n
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 equation (5.4), it results that
A(0) = FnN (0) −
1 N +1
F
(0) = 0.
n n−1
(n2 )
We conclude that for any n ≥ 1, the two-step confluent extension VeSUSY x; ω, N, −kψn k2
satisfies the enlarged shape invariant property (5.14).
Such a property is in particular satisfied if n = N = 1, as it can be directly verified. In this
case, we indeed have
(2)
L0 (z) = 1,
(1)
L1 (z) = 2 − z,
and C = 1, so that the identity (see equations (5.2) and (5.3))
1
A(z) = F11 (z) − F02 (z) = √ exp(−z) Q11 (z) − Q20 (z)
2ω


2
3
j
j
X
X
1
z 2 (2 − z) −z
d
d
2
2 
√
= √ exp(−z) 
z
−
z(2
−
z)
=
e = B(z)
dz j
dz j
2ω
2ω
j=0
j=0
is fulfilled, which implies that
2
(12 )
VeSUSY x; ω, 1, −kψ1 k2 = Ve ((0) ) x; ω, 2, −kψ0 k2 + E1 (ω).
Let us now turn ourselves to the n = 0 case. As noted above (see equation (5.7)), the formal
eigenfunction
(0
ψe0
2)
x; ω, N, −kψ0 k2 =
1
(0)
Ψ0
x; ω, N, −kψ0 k2
,
2
associated to the eigenvalue E0 = 0, is not an eigenstate of Ve (0 ) x; ω, N, −kψ0 k2 and the
(02 )
ground state of this potential is actually given by ψe
x; ω, N, −kψ0 k2 . The SUSY partner
1
therefore reads
00
2
(02 )
(02 )
VeSUSY x; ω, N, − kψ0 k2 = Ve (0 ) x; ω, N, −kψ0 k2 − 2 log ψe1
x; ω, N, −kψ0 k2
h
(02 )
i00
(0)
= V (x; ω, N ) − 2 log ψ0 (x; ω, N )Ψ0 x; ω, N, −kψ0 k2 ψe1
x; ω, N, −kψ0 k2
,
where
(0
ψe1
2)
x; ω, N, −kψ0 k2 = −E1 (ω)ψ1 (x; ω, N ) −
W (ψ0 , ψ1 | x)
(0)
Ψ0
x; ω, N, −kψ0 k2
.
24
Y. Grandati and C. Quesne
Using the shape invariance property of V , we can then write
(02 )
VeSUSY x; ω, N, −kψ0 k2 = V (x; ω, N + 1) + E1 (ω)
h
i00
(0)
− 2 log −2ωψ1 (x; ω, N )Ψ0 x; ω, N, −kψ0 k2 − W (ψ0 , ψ1 | x) ,
that is, with equation (5.8) and (5.10),
(02 )
VeSUSY x; ω, N, −kψ0 k2 = V (x; ω, N + 1) + E1 (ω) − 2[log ψ0 (x; ω, N )]00
h
i00
−(N +1/2)/2
N
− 2 log LN
(z)z
exp(−z/2)Q
(z)
−
ψ
(x;
ω,
N
+
1)
0
1
0
= V (0) (x; ω, N + 1) + E1 (ω)
n
o00
N
N +1
− 2 log z −(N +3/2)/2 exp(−z/2) LN
(z)Q
(z)
−
z
.
1
0
If there existed some constant C such that
+1
N
N +1
(z),
LN
= CQN
1 (z)Q0 (z) − z
0
(5.19)
then we would get
2
(02 )
VeSUSY x; ω, N, −kψ0 k2 = Ve (0 ) (x; ω, N + 1) + E1 (ω).
However, it can be readily seen that equation (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
2
extension Ve (0 ) .
6
Final comments
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.
As a final point, it is worth observing that apart from the Darboux (or differential) approach
to the construction of (quasi-)isospectral families of potentials, considered in the present paper,
there also exist the Abraham–Moses [1] and Pursey [40] (or integral) approaches to the same,
coming from the inverse scattering Gel’fand–Levitan or Marchenko technique. Relations between
both types of methods have been extensively studied by several authors (see, e.g., [5, 33, 38, 46]),
showing that they sometimes lead to the same results, but are in general inequivalent. The
differential procedure being more convenient and easier to use in quantum mechanics has been
preferred here. Studying the relation between our results and those of the Abraham–Moses
method would, however, be an interesting open question for future work.
References
[1] Abraham P.B., Moses H.E., Changes in potentials due to changes in the point spectrum: anharmonic
oscillators with exact solutions, Phys. Rev. A 22 (1980), 1333–1340.
Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
25
[2] Adler M., Moser J., On a class of polynomials connected with the Korteweg–de Vries equation, Comm.
Math. Phys. 61 (1978), 1–30.
[3] Adler V.È., On a modification of Crum’s method, Theoret. and Math. Phys. 101 (1994), 1381–1386.
[4] Bagchi B., Grandati Y., Quesne C., Rational extensions of the trigonometric Darboux–Pöschl–Teller potential based on para-Jacobi polynomials, J. Math. Phys. 56 (2015), 062103, 11 pages, arXiv:1411.7857.
[5] Baye D., Phase-equivalent potentials for arbitrary modifications of the bound spectrum, Phys. Rev. A 48
(1993), 2040–2047.
[6] Berezin F.A., Shubin M.A., The Schrödinger equation, Kluwer, Dordrecht, 1991.
[7] Bermudez D., Fernández C. D.J., Fernández-Garcı́a N., Wronskian differential formula for confluent supersymmetric quantum mechanics, Phys. Lett. A 376 (2012), 692–696, arXiv:1109.0079.
[8] Burchnall J.L., Chaundy T.W., A set of differential equations which can be solved by polynomials, Proc.
London Math. Soc. S2-30 (1930), 401–414.
[9] Calogero F., Yi G., Can the general solution of the second-order ODE characterizing Jacobi polynomials be
polynomial?, J. Phys. A: Math. Theor. 45 (2012), 095206, 4 pages.
[10] Cariñena J.F., Ramos A., Integrability of the Riccati equation from a group-theoretical viewpoint, Internat. J. Modern Phys. A 14 (1999), 1935–1951, math-ph/9810005.
[11] Cariñena J.F., Ramos A., Fernández C. D.J., Group theoretical approach to the intertwined Hamiltonians,
Ann. Physics 292 (2001), 42–66, math-ph/0311029.
[12] Contreras-Astorga A., Fernández C. D.J., Supersymmetric partners of the trigonometric Pöschl–Teller potentials, J. Phys. A: Math. Theor. 41 (2008), 475303, 18 pages, arXiv:0809.2760.
[13] Cooper F., Khare A., Sukhatme U., Supersymmetry in quantum mechanics, World Scientific Publishing
Co., Inc., River Edge, NJ, 2001.
[14] Crum M.M., Associated Sturm–Liouville systems, Quart. J. Math. Oxford Ser. (2) 6 (1955), 121–127,
physics/9908019.
[15] Darboux G., Sur une proposition relative aux équations linéaires, C. R. Acad. Sci. Paris 94 (1882), 1456–
1459.
[16] Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul
infinitésimal. II, 2nd ed., Gauthier-Villars, Paris, 1915.
[17] Durán A.J., Pérez M., Admissibility condition for exceptional Laguerre polynomials, J. Math. Anal. Appl.
424 (2015), 1042–1053, arXiv:1409.4901.
[18] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, McGraw-Hill,
New York, 1953.
[19] Fernández C. D.J., Salinas-Hernández E., The confluent algorithm in second-order supersymmetric quantum
mechanics, J. Phys. A: Math. Gen. 36 (2003), 2537–2543, quant-ph/0303123.
[20] Fernández C. D.J., Salinas-Hernández E., Wronskian formula for confluent second-order supersymmetric
quantum mechanics, Phys. Lett. A 338 (2005), 13–18, quant-ph/0502147.
[21] Fernández C. D.J., Salinas-Hernández E., Hyperconfluent third-order supersymmetric quantum mechanics,
J. Phys. A: Math. Theor. 44 (2011), 365302, 11 pages, arXiv:1105.2333.
[22] Gómez-Ullate D., Grandati Y., Milson R., Extended Krein–Adler theorem for the translationally shape
invariant potentials, J. Math. Phys. 55 (2014), 043510, 30 pages, arXiv:1309.3756.
[23] Gómez-Ullate D., Kamran N., Milson R., An extended class of orthogonal polynomials defined by a Sturm–
Liouville problem, J. Math. Anal. Appl. 359 (2009), 352–367, arXiv:0807.3939.
[24] Grandati Y., Solvable rational extensions of the isotonic oscillator, Ann. Physics 326 (2011), 2074–2090,
arXiv:1101.0055.
[25] Grandati Y., Multistep DBT and regular rational extensions of the isotonic oscillator, Ann. Physics 327
(2012), 2411–2431, arXiv:1108.4503.
[26] Grandati Y., New rational extensions of solvable potentials with finite bound state spectrum, Phys. Lett. A
376 (2012), 2866–2872, arXiv:1203.4149.
[27] Grandati Y., A short proof of the Gaillard–Matveev theorem based on shape invariance arguments, Phys.
Lett. A 378 (2014), 1755–1759, arXiv:1211.2392.
[28] Grandati Y., Bérard A., Rational solutions for the Riccati–Schrödinger equations associated to translationally shape invariant potentials, Ann. Physics 325 (2010), 1235–1259, arXiv:0910.4810.
26
Y. Grandati and C. Quesne
[29] Grandati Y., Bérard A., Comments on the generalized SUSY QM partnership for Darboux–Pöschl–Teller
potential and exceptional Jacobi polynomials, J. Engrg. Math. 82 (2013), 161–171.
[30] Hartman P., Ordinary differential equations, John Wiley & Sons, Inc., New York – London – Sydney, 1964.
[31] Keung W.-Y., Sukhatme U.P., Wang Q.M., Imbo T.D., Families of strictly isospectral potentials, J. Phys. A:
Math. Gen. 22 (1989), L987–L992.
[32] Krein M.G., On a continual analogue of a Christoffel formula from the theory of orthogonal polynomials,
Dokl. Akad. Nauk SSSR 113 (1957), 970–973.
[33] Luban M., Pursey D.L., New Schrödinger equations for old: inequivalence of the Darboux and Abraham–
Moses constructions, Phys. Rev. D 33 (1986), 431–436.
[34] Matveev V.B., Generalized Wronskian formula for solutions of the KdV equations: first applications, Phys.
Lett. A 166 (1992), 205–208.
[35] Matveev V.B., Positons: slowly decreasing analogues of solitons, Theoret. and Math. Phys. 131 (2002),
483–497.
[36] Messiah A., Mécanique quantique, Vol. 1, Dunod, Paris, 1959.
[37] Mielnik B., Nieto L.M., Rosas-Ortiz O., The finite difference algorithm for higher order supersymmetry,
Phys. Lett. A 269 (2000), 70–78, quant-ph/0004024.
[38] Nieto M.M., Relationship between supersymmetry and the inverse method in quantum mechanics, Phys.
Lett. B 145 (1984), 208–210.
[39] Odake S., Sasaki R., Infinitely many shape invariant potentials and new orthogonal polynomials, Phys.
Lett. B 679 (2009), 414–417, arXiv:0906.0142.
[40] Pursey D.L., New families of isospectral Hamiltonians, Phys. Rev. D 33 (1986), 1048–1055.
[41] Quesne C., Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry, J. Phys. A:
Math. Theor. 41 (2008), 392001, 6 pages, arXiv:0807.4087.
[42] Quesne C., Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum
mechanics, SIGMA 5 (2009), 084, 24 pages, arXiv:0906.2331.
[43] Quesne C., Higher-order SUSY, exactly solvable potentials, and exceptional orthogonal polynomials, Modern
Phys. Lett. A 26 (2011), 1843–1852, arXiv:1106.1990.
[44] Quesne C., Novel enlarged shape invariance property and exactly solvable rational extensions of the Rosen–
Morse II and Eckart potentials, SIGMA 8 (2012), 080, 19 pages, arXiv:1208.6165.
[45] Quesne C., Revisiting (quasi-)exactly solvable rational extensions of the Morse potential, Internat. J. Modern
Phys. A 27 (2012), 1250073, 18 pages, arXiv:1203.1811.
[46] Samsonov B.F., On the equivalence of the integral and the differential exact solution generation methods
for the one-dimensional Schrödinger equation, J. Phys. A: Math. Gen. 28 (1995), 6989–6998.
[47] Samsonov B.F., New possibilities for supersymmetry breakdown in quantum mechanics and second-order
irreducible Darboux transformations, Phys. Lett. A 263 (1999), 274–280, quant-ph/9904009.
[48] Schulze-Halberg A., Wronskian representation for confluent supersymmetric transformation chains of arbitrary order, Eur. Phys. J. Plus 128 (2013), 69, 17 pages.
[49] Sparenberg J.-M., Baye D., Supersymmetric transformations of real potentials on the line, J. Phys. A: Math.
Gen. 28 (1995), 5079–5095.
[50] Szegő G., Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, 4th ed.,
Amer. Math. Soc., Providence, R.I., 1975.
[51] Vein R., Dale P., Determinants and their applications in mathematical physics, Applied Mathematical Sciences, Vol. 134, Springer-Verlag, New York, 1999.