Multistep DBT and regular rational extensions of the isotonic oscillator
aa r X i v : . [ m a t h - ph ] N ov Multistep DBT and regular rational extensions of the isotonic oscillator
Yves Grandati
Institut de Physique, Equipe BioPhyStat, ICPMB, IF CNRS 2843,Universit´e Paul Verlaine-Metz, 1 Bd Arago, 57078 Metz, Cedex 3, France
In some recent articles we developed a new systematic approach to generate solvable rationalextensions of primary translationally shape invariant potentials. In this generalized SUSY QMpartnership, the DBT are built on the excited states Riccati-Schr¨odinger (RS) functions regularizedvia specific discrete symmetries of the considered potential. In the present paper, we prove that thisscheme can be extended in a multistep formulation. Applying this scheme to the isotonic oscillator,we obtain new towers of regular rational extensions of this potential which are strictly isospectral toit. We give explicit expressions for their eigenstates which are associated to the recently discoveredexceptional Laguerre polynomials and show explicitely that these extensions inherit of the shapeinvariance properties of the original potential.
PACS numbers:
I. INTRODUCTION
Since the pionnering work of Gomez-Ullate et al [1], the exceptional orthogonal polynomials (EOP) and their con-nexion with rational extensions of solvable quantum potentials have been an active research subject [2–20]. The EOPappear to be the constitutive elements of the eigenstates of solvable rational extensions of the second category primarytranslationally shape-invariant potentials (TSIP) [21–24] In a series of recent papers [18–20] we have developped anew approach which allows to generate infinite sets of such regular solvable extensions starting from every TSIP in avery direct and systematic way without taking recourse to any ansatz or ad hoc deforming functions. This approachis based on a generalization of the usual SUSY partnership built from excited states. The corresponding Darboux-B¨acklund Transformations (DBT), which are covariance transformations for the class of Riccati-Schr¨odinger (RS)equations [21], are based on regularized RS functions corresponding to unphysical (negative energy) eigenfunctions ofthe initial hamiltonian. They are obtained by using discrete symmetries acting on the parameters of the consideredfamily of potentials. If the use of negative energy states has been already proposed in early years of SUSY MQdevelopment [25], such a systematic scheme has never been envisaged. In the case of the isotonic oscillator, this givesthe three infinite sets L L L L L n = 2 particular case.The paper is organized as follows. First we recall the basic elements of the method in its one step form and theresults obtained for the isotonic system. The regularity of the RS functions is here obtained in a self consistentway, using disconjugacy properties of the Schr¨odinger equation [27–29] without recourse of the Kienast-Lawton-Hahntheorem [30, 31]. Then we present on a formal level the general m − step scheme and in the specific case of the isotonicoscillator, we give a criterion of concrete application in terms of asymptotic behaviour near the origin. Next weconsider in a detailed manner the particular m = 2 case, initially envisaged by Gomez-Ullate et al [26]. We then proveby induction a sufficient condition to be able to build a m − step chain of regular and strictly isospectral extendedpotentials. We finally show that the shape invariance properties of the isotonic potential are hereditary and reachedby all these extended potentials.During the writing of this article, two papers on the same subject appeared. The first one [32] contains resultsparallels to those developed here. The authors adopt a point of view similar to the one initiated in [18–20] andextended in the present paper. The so-called ”virtual states deletion method” corresponding exactly to the useof DBT based on unphysical eigenfunctions associated to regularized RS functions. The second one [33] discussesquite extensively the 2 and 3 steps cases under a slightly different angle and proposes an interesting conjecture onpossible ”degeneracies” in the set of the new generated ELP. The content of the present article has been establishedindependently of these two works. II. DARBOUX-B ¨ACKLUND TRANSFORMATIONS (DBT) AND DISCRETE SYMMETRIES If ψ λ ( x ; a ) is an eigenstate of b H ( a ) = − d /dx + V ( x ; a ) , a ∈ R m , x ∈ I ⊂ R , associated to the eigenvalue E λ ( a )( E ( a ) = 0) ψ ′′ λ ( x ; a ) + ( E λ ( a ) − V ( x ; a )) ψ λ ( x ; a ) = 0 , (1)then the Riccati-Schr¨odinger (RS) function w λ ( x ; a ) = − ψ ′ λ ( x ; a ) /ψ λ ( x ; a ) satisfies the corresponding Riccati-Schr¨odinger (RS) equation [21] − w ′ λ ( x ; a ) + w λ ( x ; a ) = V ( x ; a ) − E λ ( a ) . (2)It is a well-known fact that the set of general Riccati equations is invariant under the group G of smooth SL (2 , R )-valuedcurves M ap ( R , SL (2 , R )) [34, 35]. The particular subclass of Riccati-Schr¨odinger equations is, as for it, preserved bya specific subset of G . These transformations, called Darboux-B¨acklund Transformations (DBT), are built from anysolution w ν ( x ; a ) of the initial RS equation Eq(2) as [21, 34, 35] w λ ( x ; a ) A ( w ν ) → w ( ν ) λ ( x ; a ) = − w ν ( x ; a ) + E λ ( a ) − E ν ( a ) w ν ( x ; a ) − w λ ( x ; a ) , (3)where E λ ( a ) > E ν ( a ). w ( ν ) λ is then a solution of the RS equation: − w ( ν ) ′ λ ( x ; a ) + (cid:16) w ( ν ) λ ( x ; a ) (cid:17) = V ( ν ) ( x ; a ) − E λ ( a ) , (4)with the same energy E λ ( a ) as in Eq(2) but with a modified potential V ( ν ) ( x ; a ) = V ( x ; a ) + 2 w ′ ν ( x ; a ) . (5)This can be schematically resumed as w λ A ( w ν ) w ( ν ) λ V A ( w ν ) V ( ν ) . (6)The corresponding eigenstate of b H ( ν ) ( a ) = − d /dx + V ( ν ) ( x ; a ) can be written ψ ( ν ) λ ( x ; a ) = exp (cid:18) − Z dxw ( ν ) λ ( x ; a ) (cid:19) ∼ p E λ ( a ) − E ν ( a ) b A ( w ν ) ψ λ ( x ; a ) , (7)where b A ( a ) is a first order operator given by b A ( w ν ) = d/dx + w ν ( x ; a ) . (8)Eq(7) can still be written as ψ ( ν ) λ ( x ; a ) ∼ W ( ψ ν , ψ λ | x ) ψ ν ( x ; a ) , (9)where W ( y , y | x ) is the wronskian of the functions y , y .From V , the DBT generates a new potential V ( ν ) (quasi) isospectral to the original one and its eigenfunctions aredirectly obtained from those of V via Eq(7). Nevertheless, in general, w ν ( x ; a ) and then the transformed potential V ( ν ) ( x ; a ) are singular at the nodes of ψ ν ( x ; a ). For instance, if ψ n ( x ; a ) ( ν = n ) is a bound state of b H ( a ), V ( n ) isregular only when n = 0, that is when ψ n =0 is the ground state of b H , and we recover the usual SUSY partnership inquantum mechanics [22, 23].We can however envisage to use any other regular solution of Eq(2) as long as it has no zero on the considered realinterval I , even if it does not correspond to a physical state. For some systems, it is possible to obtain such solutionsby using specific discrete symmetries Γ i which are covariance transformations for the considered family of potentials ( a Γ i → a i V ( x ; a ) Γ i → V ( x ; a i ) = V ( x ; a ) + U ( a ) . (10)Γ i acts on the parameters of the potential and transforms the RS function of a physical excited eigenstate w n into a unphysical RS function v n,i ( x ; a ) = Γ i ( w n ( x ; a )) = w n ( x ; a i ) associated to the negative eigenvalue E n,i ( a ) =Γ i ( E n ( a )) = U ( a ) − E n ( a i ) < − v ′ n,i ( x ; a ) + v n,i ( x ; a ) = V ( x ; a ) − E n,i ( a ) . (11)To v n,i corresponds an unphysical eigenfunction of b H ( a ) φ n,i ( x ; a ) = exp (cid:18) − Z dxv n,i ( x ; a ) (cid:19) (12)associated to the eigenvalue E n,i ( a ).If the transformed RS function v n,i ( x ; a ) of Eq(11) is regular on I , it can be used to build a regular extendedpotential (see Eq(5) and Eq(7)) V ( n,i ) ( x ; a ) = V ( x ; a ) + 2 v ′ n,i ( x ; a ) (13)(quasi)isospectral to V ( x ; a ). The eigenstates of V ( n,i ) are given by (see Eq(3)) w ( n,i ) k ( x ; a ) = − v n,i ( x ; a ) + E k ( a ) −E n,i ( a ) v n,i ( x ; a ) − w k ( x ; a ) ψ ( n,i ) k ( x ; a ) = exp (cid:16) − R dxw ( n,i ) k ( x ; a ) (cid:17) ∼ √ E k ( a ) −E n,i ( a ) b A ( v n,i ) ψ k ( x ; a ) , (14)for the respective energies E k ( a ).The nature of the isospectrality depends if 1 /φ n,i ( x ; a ) satisfies or not the appropriate boundary conditions. If itis the case, then 1 /φ n,i ( x ; a ) is a physical eigenstate of b H ( n,i ) ( a ) = − d /dx + V ( n,i ) ( x ; a ) for the eigenvalue E n,i ( a )and we only have quasi-isospectrality between V ( x ; a ) and V ( n,i ) ( x ; a ). If it is not the case, the isospectrality between V ( n,i ) ( x ; a ) and V ( x ; a ) is strict.The above construction can be summarized by the following diagram E k > w k A ( v n,i ) w ( n,i ) k } Physical RS functionsΓ j ↓E k,j < v k,j A ( v n,i ) v ( n,i ) k,j } Regularized (unphysical) RS functions |{z} V A ( v n,i ) |{z} V ( n,i ) Potentials (15)with ( ψ ( n,i ) k = b A ( v n,i ) ψ k φ ( n,i ) k,j = b A ( v n,i ) φ k,j . (16)This procedure can be viewed as a ”generalized SUSY QM partnership” where the DBT can be based on excitedstates RS functions properly regularized by the symmetry Γ j . III. DISCONJUGACY AND REGULAR EXTENSIONS
To control the regularity of v n,i we can make use of the disconjugacy properties of the Schr¨odinger equation fornegative eigenvalues.A second order differential equation like Eq(1) is said to be disconjugated on I ⊂ R ( V ( x ; a ) is supposed to becontinuous on I ) if every solution of this equation has at most one zero on I [27, 28]. As it is well known, this zerois necessarily simple and at this value the considered solution changes its sign. For a closed or open interval I , thedisconjugacy of Eq(1) is equivalent to the existence of solutions of this equation which are everywhere non zero on I [27, 28]. In the following we will consider I = ]0 , + ∞ [.We have also the following result: Theorem [27, 28] If there exists a continuously differentiable solution on I of the Riccati inequation − w ′ ( x ) + w ( x ) + G ( x ) ≤ , (17)then the equation ψ ′′ ( x ) + G ( x ) ψ ( x ) = 0 (18)is disconjugated on I .In our case, since E n,i ( a ) ≤
0, we have − w ′ ( x ; a ) + w ( x ; a ) = V ( x ; a ) ≤ V ( x ; a ) − E n,i ( a ) , (19) w ( x ; a ) being continuously differentiable on I . The above theorem ensure the existence of nodeless solutions φ ( x ; a ) ofEq(1) with E λ ( a ) = E n,i ( a ), that is, of regular RS functions v ( x ; a ) solutions of Eq(11). To prove that a given solution φ ( x ; a ) belongs to this category, it is sufficient to determine the signs of the limit values φ (0 + ; a ) and φ (+ ∞ ; a ). Ifthey are identical then φ is nodeless and if they are opposite, then φ presents a unique zero on I .In the first case V ( x ; a ) + 2 v ′ ( x ; a ) constitutes a regular (quasi)isospectral extension of V ( x ; a ).In fact, Bˆocher [29] has even established a more precise theorem: Bˆocher’s disconjugacy theorem
If we can find u continuously differentiable on [ a, b ] such that u ′ ( x ) + u ( x ) < G ( x ) (20)on all [ a, b ] , then nor e ψ ( x ) = u ( x ) ψ ( x ) − ψ ′ ( x ) nor ψ ( x ) can have more than one zero in [ a, b ]. Moreover, if ψ ( x )vanish in one point of [ a, b ] then e ψ ( x ) is everywhere nonzero in this interval (and reciprocally).If this disconjugacy property is satisfied for every [ a, b ] ⊂ ] c, d [, then it is in fact satisfied in all the open interval] c, d [.Consider first the case where we have G ( x ) > , ∀ x ∈ ] c, d [. We can then take u ( x ) = 0 which implies that anysolution ψ ( x ) of Eq(18) as well as its derivative ψ ′ ( x ) have at most one zero in ] c, d [. If ξ is the corresponding zeroof ψ ′ ( x ) then ψ ( x ) is strictly monotonous on ] c, ξ [ and ] ξ , d [.If we consider now the case of Eq(1) on I = ]0 , + ∞ [ for a strictly negative eigenvalue E n,i ( a ) <
0. From Eq(19) wecan take u ( x ) = − w ( x ; a ) to apply Bˆocher’s theorem from which we deduce that not only Eq(1) is disconjugate on]0 , + ∞ [ but also the SUSY partner ψ (0) ( x ; a ) of any solution of Eq(1) can have at most one zero on ]0 , + ∞ [. Indeed(cf Eq(9)) e ψ ( x ; a ) = W ( ψ , ψ | x ) ψ ( x ; a ) = ψ (0) ( x ; a ) . (21) IV. ISOTONIC OSCILLATOR
Consider the isotonic oscillator potential (ie the radial effective potential for a three dimensional isotropic harmonicoscillator with zero ground-state energy) V ( x ; ω, a ) = ω x + a ( a − x + V ( ω, a ) , x > , (22)with a = l + 1 ≥ V ( ω, a ) = − ω (cid:0) a + (cid:1) . It is the unique exceptional primary translationally shape invariantpotential of the second category [21]. The corresponding Schr¨odinger equation is the Liouville form of the Laguerreequation on the positive half-line and its physical spectrum, associated to the asymptotic Dirichlet boundary conditions ψ (cid:0) + ; ω, a (cid:1) = 0 = ψ (+ ∞ ; ω, a ) (23)is given by ( z = ωx / , α = a − / E n ( a ) = 2 nω, ψ n ( x ; ω, a ) = x a e − z/ L αn ( z ) . (24)To ψ n corresponds the RS function [21] w n ( x ; ω, a ) = w ( x ; ω, a ) + R n ( x ; ω, a ) , (25)with w ( x ; ω, a ) = ω x − ax (26)and R n ( x ; ω, a ) = − nωωx − (2 a + 1) /x − (cid:31) ... (cid:31) n − j + 1) ωωx − (2 ( a + j ) − /x − (cid:31) ... (cid:31) ωωx − (2 ( a + n ) − /x = ωx L α +1 n − ( z ) / L αn ( z ) . (27)The shape invariance property satisfied by V is [21–24] e V ( x ; ω, a ) = V ( x ; ω, a ) + 2 w ′ ( x ; ω, a ) = V ( x ; ω, a + 1) + 2 ω (28)We have three possible discrete symmetries for V (see Eq(10)) which are given by1) ω Γ + → ( − ω ) , ( V ( x ; ω, a ) Γ + → V ( x ; ω, a ) + ω (2 a + 1) w n ( x ; ω, a ) Γ + → v n, ( x ; ω, a ) = w n ( x ; − ω, a ) , (29)2) a Γ − → − a, ( V ( x ; ω, a ) Γ − → V ( x ; ω, a ) + ω (2 a − w n ( x ; ω, a ) Γ − → v n, ( x ; ω, a ) = w n ( x ; ω, − a ) , (30)3) ( ω, a ) Γ =Γ + ◦ Γ − → ( − ω, − a ) ( V ( x ; ω, a ) Γ → V ( x ; ω, a ) + 2 ωw n ( x ; ω, a ) Γ → v n, ( x ; ω, a ) = w n ( x ; − ω, − a ) . (31)In the ( ω, α ) parameters plane, Γ + and Γ − correspond respectively to the reflections with respect to the axes ω = 0and α = 0. The RS functions v n,i , i = + , − ,
3, satisfy the respective RS equations − v ′ n,i ( x ; ω, a ) + v n,i ( x ; ω, a ) = V ( x ; ω, a ) − E n,i ( ω, a ) , (32)with E n, + ( ω, a ) = E − ( n + a +1 / ( ω ) < E n, − ( ω, a ) = E n +1 / − a ( ω ) E n, ( ω, a ) = E − ( n +1) ( ω ) < . (33)These eigenvalues are always negative in the i = + and i = 3 cases and the inequality E n, − ( ω, a ) ≤ α = a − / > n . When E n,i ( ω, a ) ≤ φ ′′ ( x ; ω, a ) + ( E n,i ( ω, a ) − V ( x ; ω, a )) φ ( x ; ω, a ) = 0 (34)is disconjugated on ]0 , + ∞ [. We are then sure that everywhere non zero solutions of Eq(34) exist, the question beingto if φ n,i ( x ; ω, a ) = exp (cid:0) − R dxv n,i ( x ; ω, a ) (cid:1) is such a solution. V. L1 AND L2 SERIES OF EXTENSIONS AND CORRESPONDING EXCEPTIONAL LAGUERREPOLYNOMIALS
The L L + and Γ − symmetries respectively in which case we havesynthetically (cid:26) E n,i ( ω, a ) = − ω ( a + i ( n + 1 / φ n,i ( x ; ω, a ) = φ ,i ( x ; ω, a ) L iαn ( − iz ) , i = ± , (35)with φ ,i ( x ; ω, a ) ∼ x iα +1 / exp ( iz/ , (36)where α is supposed to statisfy the constraint α > − in . The corresponding RS function is given by v n,i ( x ; ω, a ) = v ,i ( x ; ω, a ) + Q n,i ( x ; ω, a ) , (37)with v ,i ( x ; ω, a ) = − i ω x − iα + 1 / x = − x ( i ( z + α ) + 1 /
2) (38)and Q n,i ( x ; ω, a ) = − iωx L iα +1 n − ( − iz ) / L iαn ( − iz ) . (39)Since [30, 31] L αn ( x ) → x → + ( α +1) n n ! = (cid:0) n + αn (cid:1) L αn ( x ) ∼ x → + ∞ ( − n n ! x n , (40)where ( X ) n = ( X ) ... ( X + n −
1) is the usual Pochhammer symbol [31], we have for α > / (cid:26) φ n, + (+ ∞ ; ω, a ) = + ∞ φ n, + (0 + ; ω, a ) = 0 + , (41)and for α > n (cid:26) φ n, + (+ ∞ ; ω, a ) = 0 ± φ n, + (0 + ; ω, a ) = ±∞ , (42)with ± = ( − n .Because of the constraint α > − in , Eq(34) is disconjugated both for i = + and i = − . Combined to Eq(41)and Eq(42), this ensures that φ n,i keeps a constant strictly positive sign on ]0 , + ∞ [ which means that v n,i ( x ; ω, a ) isregular on this interval. Note that this result implies in particular that is L αn ( x ) nodeless on the negative half line andthat L − αn ( x ) has no zero on the positive half line when α > n , which is in agreement with the Kienast-Lawton-Hahntheorem [30, 31]. V ( n,i ) ( x ; ω, a ) = V ( x ; ω, a ) + 2 v ′ n,i ( x ; ω, a ) is then a regular extension of V ( x ; ω, a ). Since 1 /φ n,i ( x ; ω, a ) divergesat the origin, it cannot be a physical eigenstate of b H ( n,i ) . Consequently, V ( n,i ) and V are strictly isospectral.The (unnormalized) physical eigenstates of V ( n,i ) , ψ ( n,i ) k ( x ; ω, a ) = exp (cid:16) − R dxw ( n,i ) k ( x ; ω, a ) (cid:17) , satisfy b H ( n,i ) ( ω, a ) ψ ( n,i ) k ( x ; ω, a ) = E k ( ω ) ψ ( n,i ) k ( x ; ω, a ) − w ( n,i ) ′ k ( x ; ω, a ) + (cid:16) w ( n,i ) k ( x ; ω, a ) (cid:17) = V ( n,i ) ( x ; ω, a ) − E k ( ω ) , (43)where, using the shape invariance property of V (cf Eq(28)), we can write V ( n,i ) ( x ; ω, a ) = V ( x ; ω, a − ) + 2 Q ′ n,i ( x ; ω, a ) . (44)From Eq(14) we have also ψ ( n,i ) k ( x ; ω, a ) = b A ( v n,i ) ψ k ( x ; ω, a ) , k ≥ . (45)More precisely, using the identities [30, 31] (cid:26) L αn ( x ) + L α +1 n − ( x ) = L α +1 n ( x ) x L α +1 n − ( x ) − α L αn − ( x ) − n L α − n ( x ) = 0 , (46)we obtain ψ ( n,i ) k ( x ; ω, a ) = ( v n,i ( x ; ω, a ) − w k ( x ; ω, a )) ψ k ( x ; ω, a ) (47) ∼ L in,k,α ( z ) x a + i exp( − z/ L iαn ( − iz ) , where (cid:26) L + n,k,α ( z ) = L αn ( − z ) L α +1 k ( z ) + L α +1 n − ( − z ) L αk ( z ) L − n,k,α ( z ) = ( k + n + α ) L αk ( z ) L − αn ( z ) − ( − n + α ) L αk ( z ) L − αn − ( z ) − ( k + α ) L αk − ( z ) L − αn ( z ) , (48)are polynomials of degree n + k , namely exceptional Laguerre polynomials (ELP) of the L L n and a , we retrieve the fact that the ELP L in,k,α ( z ) constitute orthogonal families with the corresponding weights W in ( z ) = z a + i/ exp( − z )( L iαn ( − iz )) . (49) VI. FORMAL SCHEME FOR MULTI-STEP DBTA. General elements
The question of successive iterations of DBT is very natural and is at the center of the construction of the hierarchyof hamiltonians in the usual SUSY QM scheme [41]. Staying at the formal level, it can be simply described by thefollowing straightforward generalization of Eq(6) w λ A ( w ν ) w ( ν ) λ A ( w ( ν ν ) w ( ν ,ν ) λ ... A ( w ( ν ,,...,νm − ) νm ) w ( ν ,...,ν m ) λ V A ( w ν ) V ( ν ) A ( w ( ν ν ) V ( ν ,ν ) ... A ( w ( ν ,,...,νm − ) νm ) V ( ν ,...,ν m ) , (50)where w ( ν ,...,ν m ) λ is a RS function associated to the eigenvalue E λ of the potential V ( ν ,...,ν m ) ( x ; a ) = V ( x ; a ) + 2 m − X j =1 (cid:16) w ( ν ,,...,ν j − ) ν j ( x ; a ) (cid:17) ′ . (51)The corresponding eigenfunction is given by (cf Eq(7) and Eq(9)) ψ ( ν ,...,ν m ) λ ( x ; a ) = b A (cid:16) w ( ν ,...,ν m − ) ν m (cid:17) ψ ( ν ,...,ν m − ) λ ( x ; a ) = b A (cid:16) w ( ν ,...,ν m − ) ν m (cid:17) ... b A ( w ν ) ψ λ ( x ; a ) , (52)that is, ψ ( ν ,...,ν m ) λ ( x ; a ) = (cid:16) w ( ν ,...,ν m − ) ν m ( x ; a ) − w ( ν ,...,ν m − ) λ ( x ; a ) (cid:17) ψ ( ν ,...,ν m − ) λ ( x ; a ) (53)= W (cid:16) ψ ( ν ,...,ν m − ) ν m , ψ ( ν ,...,ν m − ) λ | x (cid:17) ψ ( ν ,...,ν m − ) ν m ( x ; a ) , where W ( y , ..., y m | x ) is the wronskian of the functions y , ..., y m W ( y , ..., y m | x ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ( x ) ... y m ( x ) ... ...y ( m − ( x ) ... y ( m − m ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (54)Other useful representations for the extended potentials and of their eigenfunctions are possible. From Sylvester’stheorem we can deduce [36] the following property for the Wronskians W ( y , ..., y m , y | x ) = W ( W ( y , ..., y m ) , W ( y , ..., y m − , y ) | x ) W ( y , ..., y m − | x ) . (55)Using the identity [36] W ( uy , ..., uy m | x ) = u m W ( y , ..., y m | x ) , (56)this gives W ( y , ..., y m , y | x ) W ( y , ..., y m | x ) = W (cid:16) W ( y ,...,y m ) W ( y ,...,y m − ) , W ( y ,...,y m − ,y ) W ( y ,...,y m − ) | x (cid:17) W ( y ,...,y m | x ) W ( y ,...,y m − | x ) (57)and comparing to Eq(53), we obtain ψ ( ν ,...,ν m ) λ ( x ; a ) = W ( ψ ν , ..., ψ ν m , ψ λ | x ) W ( ψ ν , ..., ψ ν m | x ) , (58)which is the well known Crum formula [37, 40] for the eigenfunctions.Inserting this result in Eq(51), we then deduce the Crum formula for the potential [37, 40] V ( ν ,...,ν m ) ( x ; a ) = V ( x ; a ) + 2 (log W ( ψ ν , ..., ψ ν m | x )) ′′ . (59)Since all the functions ψ ν j implied in the wronskians in Eq(58) and Eq(59) are eigenfunctions of the same hamil-tonian b H ( a ), the properties of the determinants allow us to replace in these wronskians the even derivatives ψ (2 l ) ν j by (cid:0) − E ν j (cid:1) l ψ ν j and the odd derivatives ψ (2 l +1) ν j by (cid:0) − E ν j (cid:1) l ψ ′ ν j .For instance W ( ψ ν , ..., ψ ν l | x ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ν ( x ; a ) ... ψ ν l ( x ; a ) ψ ′ ν ( x ; a ) ... ψ ′ ν l ( x ; a ) ... ... ( − E ν ) l − ψ ν ( x ; a ) ... ( − E ν l ) l − ψ ν l ( x ; a )( − E ν ) l − ψ ′ ν ( x ; a ) ... ( − E ν l ) l − ψ ′ ν l ( x ; a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (60)= ∆ ( ν ,...,ν l ) ( x ; a ) l Y j =1 ψ ν j ( x ; a ) , where ∆ ( ν ,...,ν l ) ( x ; a ) = ( − l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ... w ν ( x ; a ) ... w ν l ( x ; a ) ... ... ( E ν ) l − ... ( E ν l ) l − ( E ν ) l − w ν ( x ; a ) ... ( E ν l ) l − w ν l ( x ; a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (61)and W (cid:0) ψ ν , ..., ψ ν l +1 | x (cid:1) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ν ( x ; a ) ... ψ ν l ( x ; a ) ψ ′ ν ( x ; a ) ... ψ ′ ν l ( x ; a ) ... ... ( − E ν ) l − ψ ′ ν ( x ; a ) ... (cid:0) − E ν l +1 (cid:1) l − ψ ′ ν l +1 ( x ; a )( − E ν ) l ψ ν ( x ; a ) ... (cid:0) − E ν l +1 (cid:1) l ψ ν l +1 ( x ; a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (62)= ∆ ( ν ,...,ν l +1 ) ( x ; a ) l +1 Y j =1 ψ ν j ( x ; a ) , where ∆ ( ν ,...,ν l +1 ) ( x ; a ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ... w ν ( x ; a ) ... w ν l +1 ( x ; a ) ... ... ( E ν ) l − w ν ( x ; a ) ... ( E ν l ) l − w ν l +1 ( x ; a )( E ν ) l ... ( E ν l +1 ) l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (63)Then we obtain the representation of the ψ ( ν ,...,ν m ) λ in terms the so-called Crum-Krein determinants ∆ ( ν ,...,ν m ) [37, 38, 44] ψ ( ν ,...,ν m ) λ ( x ; a ) = ψ λ ( x ; a ) ∆ ( ν ,...,ν m ,λ ) ( x ; a )∆ ( ν ,...,ν m ) ( x ; a ) . (64)0Until now, all these results are purely formal. The central problem is now to choose in an appropriate way thefamily of eigenfunctions ( ψ ν , ..., ψ νm ) from which the successive DBT are built (see Eq(58) and Eq(51)) in order toensure the regularity of the successive extended potentials V ( ν ,...,ν m ) .A first answer has been given more than half a century ago by Crum [37]. The Crum proposal [37] is to take thesuccessive eigenstates of the discrete bound states spectrum starting from the ground state( ν , ..., ν m ) = (0 , ..., m −
1) (65)Krein [38, 44, 45] extended this result showing it is possible to choose sets of unnecessarily successive eigenstates ifthey satisfy a given condition, namely ( ν , ..., ν m ) = ( n , ..., n m ) ∈ N m , (66)with ( n − n )( n − n ) ... ( n − n m ) ≥ , ∀ n ∈ N . (67)This last is verified if the set ( n , ..., n m ) is constituted by ”aggregates” of an even number of eigenstates. Thepossibility to employ sets of two juxtaposed eigenstates has been rediscovered later by Adler [39]. These results havebeen used extensively in the context of higher order SUSY (see for instance [25, 41–52]). In [48], Samsonov hasstill extended the Krein-Adler result. Note finally that Fernandez et al [49] employed the specific ”Wick rotated”eigenfunctions of negative energies introduced by Shnol’ [18, 46, 53] to build successive extensions of the harmonicpotential. B. Application to the isotonic oscillator
Our aim is to consider the possibility to generalize our construction in a multi-step version on the basis of theregularized RS functions (or unphysical eigenfunctions) obtained above. Staying at a formal level, it would correspondto the following generalization of the diagram Eq(15) (see Eq(50)) E k > w k A ( v n ,i ) w ( n ,i ) k A ( v ( n ,i n ,i ) w ( n ,i ,n ,i ) k ... A ( v ( n ,i ,,...,nm − ,im − ) nm,im ) w ( n ,i ,,...,n m ,i m ) k Γ i ↓E k,i < v k,i A ( v n ,i ) v ( n ,i ) k,i A ( v ( n ,i n ,i ) v ( n ,i ,n ,i ) k,i ... A ( v ( n ,i ,,...,nm − ,im − ) nm,im ) v ( n ,i ,,...,n m ,i m ) k,i | {z } V A ( v n ,i ) | {z } V ( n ,i ) A ( v ( n ,i n ,i ) | {z } V ( n ,i ,n ,i ) ... A ( v ( n ,i ,,...,nm − ,im − ) nm,im ) | {z } V ( n ,i ,,...,n m ,i m ) (68)with v ( n ) k = v k . Under this way, we generate chains of isospectral extensions V ( n ,i ,,...,n m ,i m ) of the potential V with(see Eq(51) and Eq(59)) V ( n ,i ,...,n m ,i m ) ( x ; a ) = V ( x ; a ) + 2 (cid:16) v ( n ,i ,...,n m − ,i m − ) n m ,i m ( x ; a ) (cid:17) ′ (69)= V ( x ; a ) + 2 m X j =1 (cid:16) v ( n ,i ,...,n j − ,i j − ) n j ,i j ( x ; a ) (cid:17) ′ = V ( x ; a ) + 2 (log W ( φ n ,i , ..., φ n m ,i m | x )) ′′ w ( n ,i ,,...,n m ,i m ) k and the v ( n ,i ,,...,n m ,i m ) k,i are RS functions of the extended potential V ( n ,i ,...,n m ,i m ) associatedrespectively to the eigenvalues E k and E k,i .The corresponding eigenfunctions are given by (see Eq(53) and Eq(58))1 ψ ( n ,i ,...,n m ,i m ) k ( x ; a ) = W (cid:18) φ ( n ,i ,...,nm − ,im − ) nm,im ,ψ ( n ,i ,...,nm − ,im − ) k | x (cid:19) φ ( n ,i ,...,nm − ,im − ) nm,im ( x ; a ) = W ( φ n ,i ,...,φ nm,im ,ψ k | x ) W ( φ n ,i ,...,φ nm,im | x ) φ ( n ,i ,...,n m ,i m ) k,i ( x ; a ) = W (cid:18) φ ( n ,i ,...,nm − ,im − ) nm,im ,φ ( n ,i ,...,nm − ,im − ) k,i | x (cid:19) φ ( n ,i ,...,nm − ,im − ) nm,im ( x ; a ) = W ( φ n ,i ,...,φ nm,im ,φ k,i | x ) W ( φ n ,i ,...,φ nm,im | x ) . (70)Suppose that at the step m − V ( n ,i ,...,n m ,i m ) and of its physical eigenstates ψ ( n ,i ,,...,n m − ,i m − ) k as well as its strict isospectrality with V ( n ,i ,...,n m − ,i m − ) . This is achieved if the unphysical eigenfunction φ ( n ,i ,,...,n m − ,i m − ) n m ,i m , associated to the DBT A ( v ( n ,i ,,...,n m − ,i m − ) n m ,i m ), is nodeless and if 1 /φ ( n ,i ,,...,n m − ,i m − ) n m ,i m can-not be into the set of physical eigenstates of V ( n ,i ,...,n m − ,i m − ) .The disconjugacy of the Schr¨odinger equation φ ′′ ( x ; ω, a ) + (cid:16) E n m ,i m ( ω, a ) − V ( n ,i ,...,n m − ,i m − ) ( x ; ω, a ) (cid:17) φ ( x ; ω, a ) = 0 , (71)for a negative eigenvalue E n m ,i m guarantees that v ( n ,i ,...,n m − ,i m − ) n m ,i m is at most singular in one point. The regularityof the extended potential V ( n ,i ,...,n m ,i m ) and its strict isospectrality with the preceding one in the chain are thensatisfied as soon as the unphysical eigenfunction φ ( n ,i ,,...,n m − ,i m − ) n m ,i m satisfies appropriate boundary conditions, whichwas already the argument used above in the one step case. Namely, the strict isospectrality and the regularity aresatisfied when φ ( n ,i ,...,n m − ,i m − ) n m ,i m tends to ∞ at one extremity of ]0 , + ∞ [ and to 0 at the other extremity, with thesame sign for both limits.In the case of the isotonic oscillator, there are in fact three type of regularization transformations Γ i , i = + , − , L L L L L L φ ( n ,i ,...,n m − ,i m − ) n m ,i m ( x ; ω, a ) = φ ,i m ( x ; ω, a ) R ( n ,i ,,...,n m ,i m ) ( x ; ω, a ) , (72)where R ( n ,i ,,...,n m ,i m ) ( x ; ω, a ) = L i m αn m ( − i m z ) ∆ ( n ,i ,,...,n m ,i m ) ( x ; ω, a )∆ ( n ,i ,,...,n m − ,i m − ) ( x ; ω, a ) (73)is a rational function. Due to the presence of the exponential in φ ,i m , the behaviour at infinity of φ ( n ,i ,...,n m − ,i m − ) n m ,i m is the same as for φ ,i m , namely φ ( n ,i ,...,n m − ,i m − ) n m , + ( x ; ω, a ) ∼ x → + ∞ exp ( z/ x α +1 / R ( n ,i ,,...,n m , +) (+ ∞ ; ω, a ) → x → + ∞ ±∞ φ ( n ,i ,...,n m − ,i m − ) n m , − ( x ; ω, a ) ∼ x → + ∞ exp ( − z/ x − α +1 / R ( n ,i ,,...,n m , − ) (+ ∞ ; ω, a ) → x → + ∞ ± , (74)(the sign of the limit being the one of R ( n ,i ,,...,n m , +) (+ ∞ ; ω, a ) or R ( n ,i ,,...,n m , − ) (+ ∞ ; ω, a ) respectively).The absence of node for φ ( n ,i ,...,n m − ,i m − ) n m ,i m and the strict isospectrality between V ( n ,i ,...,n m − ,i m − ) and V ( n ,i ,...,n m ,i m ) , are simultaneously ensured if and only if we have φ ( n ,i ,...,n m − ,i m − ) n m , + ( x ; ω, a ) ∼ x → + x α +1 / R ( n ,i ,,...,n m , +) (0 + ; ω, a ) → x → + ± φ ( n ,i ,...,n m − ,i m − ) n m , − ( x ; ω, a ) ∼ x → + x − α +1 / R ( n ,i ,,...,n m , − ) (0 + ; ω, a ) → x → + ±∞ , (75)(the sign of the limit being the one of R ( n ,i ,,...,n m , +) (0 + ; ω, a ) or R ( n ,i ,,...,n m , − ) (0 + ; ω, a ) respectively), with sign (cid:16) R ( n ,i ,,...,n m ,i m ) (cid:0) + ; ω, a (cid:1)(cid:17) = sign (cid:16) R ( n ,i ,,...,n m ,i m ) (+ ∞ ; ω, a ) (cid:17) . VII. TWO-STEP CASE
Before to consider the general case, we first consider the two-step case. We then have to check that the unphysicaleigenstates φ ( n ,i ) n ,i satisfy the appropriate boundary conditions. Using Eq(35) in Eq(70), we have for m = 2 φ ( n ,i ) n ,i ( x ; ω, a ) = W (cid:0) φ ,i ( x ; ω, a ) L i αn ( − i z ) , φ ,i ( x ; ω, a ) L i αn ( − i z ) | x (cid:1) φ ,i ( x ; ω, a ) L i αn ( − i z ) , where α > − i j n j , j = 1 ,
2, or with Eq(56) φ ( n ,i ) n ,i ( x ; ω, a ) = φ ,i ( x ; ω, a ) W (cid:16) L i αn ( − i z ) , ( x α exp ( z/ i − i L i αn ( − i z ) | x (cid:17) L i αn ( − i z ) . (76)If i = i = i , then φ ( n ,i ) n ,i ( x ; ω, a ) = φ ,i ( x ; ω, a ) W (cid:0) L iαn ( − iz ) , L iαn ( − iz ) | x (cid:1) L iαn ( − iz ) . (77)Using [36] W ( y , ..., y m | x ) = (cid:18) dzdx (cid:19) m ( m − / W ( y , ..., y m | z ) (78)and [30, 31] ( L αn ( x )) ′ = − L α +1 n − ( x ) , (79)this gives φ ( n ,i ) n ,i ( x ; ω, a ) = iωx iα +3 / exp ( iz/ L iαn ( − iz ) (cid:12)(cid:12)(cid:12)(cid:12) L iαn ( − iz ) L iαn ( − iz ) L iα +1 n − ( − iz ) L iα +1 n − ( − iz ) (cid:12)(cid:12)(cid:12)(cid:12) . (80)We then have (cf Eq(40)) φ ( n ,i ) n ,i ( x ; ω, a ) ∼ x → + iωx iα +3 / ( n iαn ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) n + iαn (cid:1) (cid:0) n + iαn (cid:1)(cid:0) n + iαn − (cid:1) (cid:0) n + iαn − (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = iωx iα +3 / (cid:0) n + iαn − (cid:1) n − n n φ ( n ,i ) n ,i ( x ; ω, a ) ∼ x → + ∞ i n ( n − n ) ωn ! x n + iα − / exp ( iz/ , (81)that is, φ ( n , +) n , + ( x ; ω, a ) ∼ x → + ωx ( α +2) − / (cid:0) n + αn − (cid:1) n − n n → x → + ± φ ( n , − ) n , − ( x ; ω, a ) ∼ x → + ∞ ω ( n − n ) n ! x n + α − / exp ( z/ → x → + ∞ ±∞ , , if a > , (82)where ± = sign ( n − n ), and φ ( n , − ) n , − ( x ; ω, a ) ∼ x → + ( − n ωx − ( α − − / (cid:0) α − n n − (cid:1) n − n n → x → + ±∞ φ ( n , − ) n , − ( x ; ω, a ) ∼ x → + ∞ ( − n ω ( n − n ) n ! x n − α − / exp ( − z/ → x → + ∞ ± , if a > sup ( n , n ) , (83)where ± = ( − n sign ( n − n ).3If i = − i = − i , then using the last of the following identities [54] ( z α L αn ( z )) ′ = ( n + α ) z α − L α − n ( z )( e − z L αn ( z )) ′ = − e − z L α +1 n ( z )( z α e − z L αn ( z )) ′ = ( n + 1) z α − e − z L α − n +1 ( z ) , (84)we obtain φ ( n , − i ) n ,i ( x ; ω, a ) = ωx − iα +3 / exp ( − iz/ (cid:18) ω (cid:19) iα W (cid:0) L − iαn ( iz ) , z iα exp ( iz ) L iαn ( − iz ) | z (cid:1) L − iαn ( iz ) (85)= 2 x iα − / exp ( iz/ L − iαn ( iz ) (cid:12)(cid:12)(cid:12)(cid:12) L − iαn ( iz ) z L iαn ( − iz ) − i L − iα +1 n − ( iz ) ( n + 1) L iα − n ( − iz ) (cid:12)(cid:12)(cid:12)(cid:12) . We then have (cf Eq(40)) φ ( n , − i ) n ,i ( x ; ω, a ) ∼ x → + x iα − / i n +1 ( α + in ) ...αn ! φ ( n , − i ) n ,i ( x ; ω, a ) ∼ x → + ∞ i n +1 1 n ! (cid:0) ω (cid:1) n +1 x n + iα +3 / exp ( iz/ . (86)Consequently ( a > φ ( n , − ) n , + ( x ; ω, a ) ∼ x → + x α − / α + n ) ...αn ! → x → + + φ ( n , − ) n , + ( x ; ω, a ) ∼ x → + ∞ n ! (cid:0) ω (cid:1) n +1 x n + a +1 exp ( z/ → x → + ∞ + ∞ (87)and when α > n φ ( n , +) n , − ( x ; ω, a ) ∼ x → + ( − n +1 ( α − n ) ...αn ! x − α − / → x → + ±∞ φ ( n , +) n , − ( x ; ω, a ) ∼ x → + ∞ ( − n +1 1 n ! (cid:0) ω (cid:1) n +1 x n − a +2 exp ( − z/ → x → + ∞ ± , (88)with ± = ( − n +1 .When a > sup( n , n ), we have the correct asymptotic behaviour which ensures the regularity of V ( n ,i ,n ,i ) ( x ; ω, a )and V ( n ,i ) ( x ; ω, a ) as their strict isospectrality with V ( x ; ω, a ) for every choice of ( n , i , n , i ) with n = n .We then obtain a chain with two strictly isospectral regular successive extensions of VV A ( v n ,i ) V ( n ,i ) A ( v ( n ,i n ,i ) V ( n ,i ,n ,i ) . (89) VIII. GENERAL M-STEP CASE
Consider now the general case of a m -step DBT. We proceed by induction, supposing that we have built a chain of m − V A ( v n ,i ) V ( n ,i ) A ( v ( n ,i n ,i ) ... A ( v ( n ,i ,,...,nm − ,im − ) nm − ,im − ) V ( n ,i ,...,n m − ,i m − ) . (90)and that we have α > − i j n j , ∀ j ≤ m . This implies in particular that the φ ( n ,i ,...,n j − ,i j − ) n j ,i j ( x ; ω, a ) , j ≤ m − b H ( n ,i ,...,n m − ,i m − ) associated to the respective negative eigenvalues E n j − ,i j − ( ω, a ) < , + ∞ [ and that they keep a constant sign on this interval that we can always take as positive.4We have to determine what is the constraint to implement for the unphysical eigenfunction φ ( n ,i ,...,n m − ,i m − ) n m ,i m inorder that it satisfies the good boundary conditions necessary to build a regular and strictly isospectral extension V ( n ,i ,...,n m ,i m ) at the next step.From Eq(74) we know that φ ( n ,i ,...,n m − ,i m − ) n m − , + ( x ; ω, a ) → x → + ∞ + ∞ φ ( n ,i ,...,n m − ,i m − ) n m − , − ( x ; ω, a ) → x → + ∞ + , (91)and consequently we have also (cf Eq(75)) φ ( n ,i ,...,n m − ,i m − ) n m − , + ( x ; ω, a ) → x → + + φ ( n ,i ,...,n m − ,i m − ) n m − , − ( x ; ω, a ) → x → + + ∞ . (92)For x sufficiently large or in the vinicity of 0, we have E n m − , ± ( ω, a ) < V ( n ,i ,...,n m − ,i m − ) ( x ; ω, a ). The Bˆocherdisconjugacy theorem then implies that in the neighbourhoods of 0 and of + ∞ , the functions φ ( n ,i ,...,n m − ,i m − ) n m − ,i m − arestrictly monotonous, that is strictly decreasing if i m − = − and strictly increasing if i m − = +.We already know (cf Eq(74)) that φ ( n ,i ,...,n m − ,i m − ) n m ,i m (+ ∞ ; ω, a ) = (cid:26) ±∞ , if i m = +0 ± , if i m = − (93)We prove now by induction the following result for the unphysical eigenfunctions of the potentials obtained at the( m − th step ( z = ωx / φ ( n ,i ,...,n m − ,i m − ) n m ,i m ( x ; ω, a ) ∼ x → + e φ ( n ,i ,...,n m − ,i m − ) n m ,i m (0) x i m ( α + q ( i ,...,i m )) − / (1 + O ( z )) , (94)where e φ ( n ,i ,...,n m − ,i m − ) n m ,i m (0) has the same sign as φ ( n ,i ,...,n m − ,i m − ) n m ,i m (+ ∞ ; ω, a ) and where q ( i , ..., i m ) = q + ( i , ..., i m ) − q − ( i , ..., i m ), q ± ( i , ..., i m ) is the number of i j equal to ± in the set ( i , ..., i m ). q ± ( i , ..., i m ) can beviewed as the number of state of ” ± charge” used in the chain of DBT at the m th step and the difference q ( i , ..., i m )as the total charge associated to the chain.We then have φ ( n ,i ,...,n m − ,i m − ) n m , + ( x ; ω, a ) → x → + ± , if a > − q ( i , ..., i m − , +) + 1 = − q ( i , ..., i m − ) φ ( n ,i ,...,n m − ,i m − ) n m , − ( x ; ω, a ) → x → + ±∞ , if a > − q ( i , ..., i m − , − ) = 1 − q ( i , ..., i m − ) , (95)which are necessary conditions in order that the potentials V ( n ,i ,...,n m , +) and V ( n ,i ,...,n m , − ) respectively are regularand strictly isospectral to V ( n ,i ,,...,n m − ,i m − ) . Note that these constraints are independent of the n j but dependonly on the type of the states used to build the chain.Refering to the results obtained in the 2-step case (see Eq(82), Eq(83) and Eq(88)), we can verify that this propertyis valid for m = 2 since (cid:26) q (+ , +) = − q ( − , − ) = 2 q (+ , − ) = q ( − , +) = 0 . (96)Suppose that Eq(94) is verified till the ( m − th step, that is, φ ( n ,i ,...,n m − ,i m − ) n, ± ( x ; ω, a ) ∼ x → + x ± ( α + q ( i ,...,i m − ))+1 / (1 + O ( z )) , (97)with a > − q ( i , ..., i m − , +) = − − q ( i , ..., i m − ), if i m − = +, and a > − q ( i , ..., i m − , − ) = 2 − q ( i , ..., i m − ),if i m − = − .5Since φ ( n ,i ,...,n m − ,i m − ) n m − ,i m − and φ ( n ,i ,...,n m − ,i m − ) k,i are both eigenfunctions of b H ( n ,i ,...,n m − ,i m − ) , using the wron-skian theorem [55] in Eq(70) we can write the unphysical eigenfunctions at the ( m − th step as φ ( n ,i ,...,n m − ,i m − ) k,i ( x ; ω, a ) = (cid:0) E n m − ,i m − − E k,i (cid:1) R xx dξφ ( n ,i ,...,n m − ,i m − ) n m − ,i m − ( ξ ; ω, a ) φ ( n ,i ,...,n m − ,i m − ) k,i ( ξ ; ω, a ) φ ( n ,i ,...,n m − ,i m − ) n m − ,i m − ( x ; ω, a ) (98)+ W (cid:16) φ ( n ,i ,...,n m − ,i m − ) n m − ,i m − , φ ( n ,i ,...,n m − ,i m − ) k,i | x (cid:17) φ ( n ,i ,...,n m − ,i m − ) n m − ,i m − ( x ; ω, a ) . We have to consider several cases corresponding to the four possible couples ( i m − , i ).1) Consider first the case i = i m − = + in which we take x = 0. Using Eq(92), Eq(98) gives φ ( n ,i ,...,n m − ,i m − ,n m − , +) k, + ( x ; ω, a ) ∼ R x dξφ ( n ,i ,...,n m − ,i m − ) n m − , + ( ξ ; ω, a ) φ ( n ,i ,...,n m − ,i m − ) k, + ( ξ ; ω, a ) φ ( n ,i ,...,n m − ,i m − ) n m − , + ( x ; ω, a ) . (99)Since φ ( n ,i ,...,n m − ,i m − ) n m − , + and φ ( n ,i ,...,n m − ,i m − ) k, + are both strictly positive on ]0 , + ∞ [, φ ( n ,i ,...,n m − ,i m − ,n m − , +) k, + keeps a constant sign on this interval and we can always take it as positive. Using Eq(97), Eq(99) gives φ ( n ,i ,...,n m − ,i m − ,n m − , +) k, + ( x ; ω, a ) ∼ x → + x ( α + q ( i ,...,i m − , + , +)) − / (1 + O ( z )) , (100)Consequently, in this case, Eq(94) is also verified at the m th step.2) In the case i = i m − = − , if we take x = + ∞ , we have similarly with Eq(91) and Eq(98) φ ( n ,i ,...,n m − ,i m − ,n m − , − ) k, − ( x ; ω, a ) ∼ R + ∞ x dξφ ( n ,i ,...,n m − ,i m − ) n m − , − ( ξ ; ω, a ) φ ( n ,i ,...,n m − ,i m − ) k, − ( ξ ; ω, a ) φ ( n ,i ,...,n m − ,i m − ) n m − , − ( x ; ω, a ) . (101)As in the preceding case, since φ ( n ,i ,...,n m − ,i m − ) n m − , − and φ ( n ,i ,...,n m − ,i m − ) k, − are both strictly positive on ]0 , + ∞ [, φ ( n ,i ,...,n m − ,i m − ,n m − , − ) k, − keeps a constant sign on this interval and we can always take it as positive. From Eq(74)we deduce that for any ε > R + ∞ ε dξφ ( n ,i ,...,n m − ,i m − ) n m − , − ( ξ ; ω, a ) φ ( n ,i ,...,n m − ,i m − ) k, − ( ξ ; ω, a ) is finite and positive.Since ( a > − q ( i , ..., i m − )) φ ( n ,i ,...,n m − ,i m − ) n m − , − ( x ; ω, a ) φ ( n ,i ,...,n m − ,i m − ) k, − ( x ; ω, a ) ∼ x → + x − α + q ( i ,...,i m − ))+1 (1 + O ( z )) , Eq(101) gives φ ( n ,i ,...,n m − ,i m − ,n m − , − ) k, − ( x ; ω, a ) ∼ x → + x − ( α + q ( i ,...,i m − , − , − )) − / . Eq(94) is again verified at the m th step.3) It remains to consider the cases i = + = − i m − and i = − = − i m − . Since φ ( n ,i ,...,n m − ,i m − ) n, + and φ ( n ,i ,...,n m − ,i m − ) n, − have not the same behavior nor at 0 nor at + ∞ , Eq(98) is not adapted and we proceed to adirect evaluation of the wronskian W (cid:16) φ ( n ,i ,...,n m − ,i m − ) n m − , − , φ ( n ,i ,...,n m − ,i m − ) k, + | x (cid:17) .Since φ ( n ,i ,...,n m − ,i m − ) n m − , − and φ ( n ,i ,...,n m − ,i m − ) k, + are strictly positive and respectively decreasing and increasingboth at infinity and in the vinicity of the origin, W (cid:16) φ ( n ,i ,...,n m − ,i m − ) n m − , − , φ ( n ,i ,...,n m − ,i m − ) k, + | x (cid:17) has the same positivesign at these two limits. Due to the disconjugacy of φ ′′ ( x ; ω, a ) + (cid:16) E k, + ( ω, a ) − V ( n ,i ,...,n m − ,i m − ) ( x ; ω, a ) (cid:17) φ ( x ; ω, a ) = 0 , (102)6we then deduce that φ ( n ,i ,...,n m − ,i m − ,n m − , − ) k, + ( x ; ω, a ) keeps a constant sign (taken as positive) on all the interval]0 , + ∞ [.Using (see Eq(97)) φ ( n ,i ,...,n m − ,i m − ) k, + ( x ; ω, a ) ∼ x → + x ( α + q ( i ,...,i m − ))+1 / (cid:16) e φ ( n ,i ,...,n m − ,i m − ) k, + (0) + O ( z ) (cid:17) φ ( n ,i ,...,n m − ,i m − ) n m − , − ( x ; ω, a ) ∼ x → + x − ( α + q ( i ,...,i m − ))+1 / (cid:16) e φ ( n ,i ,...,n m − ,i m − ) n m − , − (0) + O ( z ) (cid:17) , (103)we obtain W (cid:16) φ ( n ,i ,...,n m − ,i m − ) n m − , − , φ ( n ,i ,...,n m − ,i m − ) k, + | x (cid:17) ∼ x → + α + q ( i , ..., i m − )) e φ ( n ,i ,...,n m − ,i m − ) k, + (0) e φ ( n ,i ,...,n m − ,i m − ) n m , − (0) + O ( z ) . (104)Consequently, we deduce φ ( n ,i ,...,n m − ,i m − ,n m − , − ) k, + ( x ; ω, a ) ∼ x → + x ( α + q ( i ,...,i m − , − , +)) − / (105)and φ ( n ,i ,...,n m − ,i m − ,n m − , +) k, − ( x ; ω, a ) = W (cid:16) φ ( n ,i ,...,n m − ,i m − ) n m − , + , φ ( n ,i ,...,n m − ,i m − ) k, − | x (cid:17) φ ( n ,i ,...,n m − ,i m − ) n m , + ( x ; ω, a ) (106) ∼ x → + − α + q ( i , ..., i m − )) e φ ( n ,i ,...,n m − ,i m − ) k, − (0) x − ( α + q ( i ,...,i m − )) − / (1 + O ( z )) ∼ x → + x − ( α + q ( i ,...,i m − , + , − )) − / . The result Eq(94) is then still valid at the m th step.We still have to verify that this condition is compatible with the regularity of the preceding potentials in thechain. For that, note that at the m th step the explicit expression of the resulting potential and its associatedeigenfunctions are a priori independent of the ordering chosen for the sequence . This means that we can con-sider several chains leading to the same extended potential. Consider the set ( n , i , ..., n m − , i m − , n m − , i m − )in which we have regrouped first all the n j associated to a positive i j , that is, we have re-arranged the chain as (cid:16) n ′ , i ′ , ..., n ′ q + , i ′ q + , n ′ q + +1 , i ′ q + +1 , ..., n ′ m − , i ′ m − (cid:17) = (cid:16) n ′ , + , ..., n ′ q + , + , n ′ q + +1 , − , ..., n ′ m − , − (cid:17) , where (cid:0) n ′ , ..., n ′ m − (cid:1) isa permutation of ( n , ..., n m − ) and where q + = q + ( i , ..., i m − ).Suppose first that q + = m −
1, that is, i ′ j = + , ∀ j ∈ { , ..., m − } . ∀ j ∈ { , ..., m − } we have 1 − q + (cid:0) i ′ , ..., i ′ j (cid:1) ≤ a >
1, the φ ( n ′ ,i ′ ,...,n ′ j ,i ′ j ) n ′ j +1 ,i ′ j +1 are all nodeless implying the regularity of all the potentials V ( n ′ ,i ′ ,...,n ′ j ,i ′ j ) in thechain.Suppose now that q + < m −
1, that is, i ′ m − = − . The condition a > − q (cid:0) i ′ , ..., i ′ m − (cid:1) = 2 − q (cid:0) i ′ , ..., i ′ m − (cid:1) > − q (cid:0) i ′ , ..., i ′ m − (cid:1) implies the absence of node for φ ( n ′ ,i ′ ,...,n ′ m − ,i ′ m − ) n ′ m − , − and consequently the regularityof V ( n ′ ,i ′ ,...,n ′ m − ,i ′ m − ).We can iterate this reasoning to go backward along the chain and we see clearly that at each step, the opposite ofthe ”total charge” decreases, ensuring the regularity of the associated potential. In this procedure, once we attain thelevel q + , we have eliminated all the negative i ′ j and retrieve then the first case.Consequently, if we satisfy the condition a > − q ( i , ..., i m − ) at the step m , we are ensured of the existence of achain of regular extensions which leads to a regular V ( n ,i ,,...,n m ,i m ) . IX. EIGENSTATES OF THE EXTENDED POTENTIALS
We can generalize the calculations made in the two-step case to determine explicit expressions for the eigenstates ofthe successive extensions. For that we adopt the ordering mentioned above and consider the spectrum of the potential V ( n , + ,...,n q + , + ,n q ++1 , − ,....,n m , − ) where q + = q + ( i , ..., i m ). Using Eq(56) and Eq(78), we have7 W (cid:16) φ n , + , ..., φ n q + , + , ..., φ n q ++1 , − , ..., φ n m , − , ψ k | x (cid:17) (107) ∼ x ( m +1)( m/ a ) e − ( m +1) z/ W (cid:16) e z L αn ( − z ) , ..., e z L αn q + ( − z ) , z − α L − αn q ++1 ( z ) , ..., z − α L − αn m ( z ) , L αk ( z ) | x (cid:17) , that is, with Eq(79) and Eq(84) W (cid:16) φ n , + , ..., φ n q + , + , ..., φ n q ++1 , − , ..., φ n m , − , ψ k | x (cid:17) (108) ∼ x ( m +1)( m/ a ) e ( q + − ( m +1) / z z − ( m − q + )( α + m ) det (cid:18) Ψ( n , + ,...,n q + , + ,n q ++1 , − ,....,n m , − ) k (cid:19) , where ( i, j ∈ { , ..., m + 1 } ) (cid:20) Ψ( n , + ,...,n q + , + ,n q ++1 , − ,....,n m , − ) k (cid:21) i,j = L α + i − n j ( − z ) , if j ≤ q + ( n j − α − i + 2) i − z m − i +1 L − α − i +1 n j ( z ) , if q + + 1 ≤ j ≤ m ( − i − L α + i − k − i +1 ( z ) , if j = m + 1 . (109)In the same manner W (cid:16) φ n , + , ..., φ n q + , + , ..., φ n q ++1 , − , ..., φ n m , − | x (cid:17) (110) ∼ x m (( m − / a ) e mz/ W (cid:16) L αn ( − z ) , ..., L αn q + ( − z ) , z − α e − z L − αn q ++1 ( z ) , ..., z − α e − z L − αn m ( z ) | x (cid:17) , that is, W (cid:16) φ n , + , ..., φ n q + , + , ..., φ n q ++1 , − , ..., φ n m , − | x (cid:17) (111) ∼ x m (( m − / a ) z ( m − q + )( − α − m +1)) e − ( m/ − q + ) z det (cid:16) Φ( n , + ,...,n q + , + ,n q ++1 , − ,....,n m , − ) (cid:17) , where ( i, j ∈ { , ..., m } ) h Φ( n , + ,...,n q + , + ,n q ++1 , − ,....,n m , − ) i i,j = ( L α + i − n j − i +1 ( − z ) , if j ≤ q + ( n j + 1) i − z m − i +1 L − α − i +1 n j + i − ( z ) , if q + + 1 ≤ j ≤ m. (112)Eq(70) gives then ψ ( n , + ,...,n q + , + ,n q ++1 , − ,....,n m , − ) k ( x ; ω, a ) ∼ ψ ( x ; ω, a + q ( i , ..., i m )) det (cid:18) Ψ( n , + ,...,n q + , + ,n q ++1 , − ,....,n m , − ) k (cid:19) det (cid:16) Φ( n , + ,...,n q + , + ,n q ++1 , − ,....,n m , − ) (cid:17) . (113)The ψ ( n ,i ,...,n m ,i m ) k are then obtained as the product of the modified ”gauge factor” (ie fundamental state ofthe initial potential) ψ ( x ; ω, a + q ( i , ..., i m )), multiplied by the ratio of two polynomials, the polynomial denom-inator det (cid:16) Φ( n , + ,...,n q + , + ,n q ++1 , − ,....,n m , − ) (cid:17) being common to all the eigenstates of the considered extension. Theorthogonality condition on the ψ ( n ,i ,...,n m ,i m ) k ensure that the det (cid:18) Ψ( n , + ,...,n q + , + ,n q ++1 , − ,....,n m , − ) k (cid:19) constitute anorthogonal family with respect to the weight W ( n ,i ,...,n m ,i m ) ( z ) = z α + q ( i ,...,i m ) e − z (cid:16) det (cid:16) Φ( n , + ,...,n q + , + ,n q ++1 , − ,....,n m , − ) (cid:17)(cid:17) . X. SHAPE INVARIANCE OF THE EXTENDED POTENTIALS
Consider a chain of regular strictly isospectral extensions V A ( v n ,i ) V ( n ,i ) A ( v ( n ,i n ,i ) ... A ( v ( n ,i ,,...,nm − ,im − ) nm,im ) V ( n ,i ,,...,n m ,i m ) , (114)The superpartner of the potential V ( n ,i ,...,n m ,i m ) ( x ; ω, a ) is given by [22, 23] e V ( n ,i ,...,n m ,i m ) ( x ; ω, a ) = V ( n ,i ,...,n m ,i m ) ( x ; ω, a ) + 2 (cid:16) w ( n ,i ,...,n m ,i m )0 ( x ; ω, a ) (cid:17) ′ , (115) w ( n ,i ,...,n m ,i m )0 ( x ; ω, a ) being the RS function associated to the ground level of V ( n ,i ,...,n m ,i m ) ( E ( ω ) = 0).Since (see Eq(3)) w ( n ,i ,...,n m ,i m )0 ( x ; ω, a ) = − v ( n ,i ,...,n m − ,i m − ) n m ,i m ( x ; ω, a ) − E n m ,i m ( ω, a ) v ( n ,i ,...,n m − ,i m − ) n m ,i m ( x ; ω, a ) − w ( n ,i ,...,n m − ,i m − )0 ( x ; ω, a ) , (116)we have with Eq(69) e V ( n ,i ,...,n m ,i m ) ( x ; ω, a ) = V ( n ,i ,...,n m − ,i m − ) ( x ; ω, a ) (117) − E n m ,i m ( ω, a ) v ( n ,i ,...,n m − ,i m − ) n m ,i m ( x ; ω, a ) − w ( n ,i ,...,n m − ,i m − )0 ( x ; ω, a ) ! ′ . (118)We proceed by induction. Suppose that the potential V ( n ,i ,...,n m − ,i m − ) has the same shape invariance propertiesthan V e V ( n ,i ,...,n m − ,i m − ) ( x ; ω, a ) = V ( n ,i ,...,n m − ,i m − ) ( x ; ω, a ) + 2 (cid:16) w ( n ,i ,...,n m − ,i m − )0 ( x ; ω, a ) (cid:17) ′ (119)= V ( n ,i ,...,n m − ,i m − ) ( x ; ω, a + 1) + 2 ω. As shown in [19], this is effectively the case for m = 2. It results e V ( n ,i ,...,n m ,i m ) ( x ; ω, a ) = V ( n ,i ,...,n m ,i m ) ( x ; ω, a + 1) + 2 ω − (cid:16) ∆ ( n ,i ,...,n m ,i m ) ( x ; ω, a ) (cid:17) ′ , (120)where ∆ ( n ,i ,...,n m ,i m ) ( x ; ω, a ) = E n m ,i m ( ω, a ) v ( n ,i ,...,n m − ,i m − ) n m ,i m ( x ; ω, a ) − w ( n ,i ,...,n m − ,i m − )0 ( x ; ω, a ) (121)+ w ( n ,i ,...,n m − ,i m − )0 ( x ; ω, a ) + v ( n ,i ,...,n m − ,i m − ) n m ,i m ( x ; ω, a + 1) . Suppose also that ∆ ( n ,i ,...,n m − ,i m − ) ( x ; ω, a ) = 0 , (122)which again is verified for m = 2 [19]. We can then write v ( n ,i ,...,n m − ,i m − ) n m − ,i m − ( x ; ω, a + 1) = − w ( n ,i ,...,n m − ,i m − )0 ( x ; ω, a ) (123) − E n m − ,i m − ( ω, a ) v ( n ,i ,...,n m − ,i m − ) n m − ,i m − ( x ; ω, a ) − w ( n ,i ,...,n m − ,i m − )0 ( x ; ω, a ) . w ( n ,i ,...,n m − ,i m − )0 ( x ; ω, a ) = − v ( n ,i ,...,n m − ,i m − ) n m − ,i m − ( x ; ω, a ) − E nm − ,im − ( ω,a ) v ( n ,i ,...,nm − ,im − ) nm − ,im − ( x ; ω,a ) − w ( n ,i ,...,nm − ,im − ) ( x ; ω,a ) v ( n ,i ,...,n m − ,i m − ) n m ,i m ( x ; ω, a ) = − v ( n ,i ,...,n m − ,i m − ) n m − ,i m − ( x ; ω, a ) + E nm,im ( ω,a ) −E nm − ,im − ( ω,a ) v ( n ,i ,...,nm − ,im − ) nm − ,im − ( x ; ω,a ) − v ( n ,i ,...,nm − ,im − ) nm,im ( x ; ω,a ) (124)and (see Eq(33)) E n m ,i m ( ω, a + 1) − E n m − ,i m − ( ω, a + 1) = E n m ,i m ( ω, a ) − E n m − ,i m − ( ω, a ) . (125)Inserting Eq(124), Eq(123) and Eq(125) into Eq(121), we obtain after a little elementary algebra∆ ( n ,i ,...,n m ,i m ) ( x ; ω, a ) = 0 . (126)By induction, this property is verified for every m . Then Eq(120) becomes e V ( n ,i ,...,n m ,i m ) ( x ; ω, a ) = e V ( n ,i ,...,n m ,i m ) ( x ; ω, a + 1) + 2 ω, (127)that is, the potential has also the same shape invariance properties than the isotonic potential. The translationalshape invariance of the isotonic potential is then hereditary in such chain of extensions. XI. CONCLUSION
We have shown that the ”generalized SUSY QM partnership” that we have previously elaborated in a one-stepscheme can be extended in a multi-step formulation. We have proven the necessary conditions to obtain chains ofregular extensions of the isotonic potential of arbitrary length and have given explicit expressions for their eigenstates.We also established explicitely the hereditary character of the shape invariance properties of the isotonic potentialwhich are common to all the potentials in a given chain.The case of the generic potentials of the second category of primary TSIP [21], namely P¨oschl-Teller or Scarfpotentials, can be considered in the same way. This work is in progress and a forthcoming paper is in preparation.For the first category exceptional TSIP, due to the strict isospectrality constraint, the only interesting case in whichwe can envisage such chain of extensions is the ERKC potential [20]. This is the object of further investigations.
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