Position Dependent Mass Schroedinger Equation and Isospectral Potentials : Intertwining Operator approach
aa r X i v : . [ m a t h - ph ] F e b Position Dependent Mass Schr¨odinger Equation and Isospectral Potentials :Intertwining Operator approach
Bikashkali Midya, ∗ B. Roy, † and R. Roychoudhury ‡ Physics & Applied Mathematics UnitIndian Statistical InstituteKolkata 700108India
Here we have studied first and second-order intertwining approach to generate isospectral partnerpotentials of position-dependent (effective) mass Schr¨odinger equation. The second-order intertwineris constructed directly by taking it as second order linear differential operator with position depndentcoefficients and the system of equations arising from the intertwining relationship is solved for thecoefficients by taking an ansatz. A complete scheme for obtaining general solution is obtained whichis valid for any arbitrary potential and mass function. The proposed technique allows us to generateisospectral potentials with the following spectral modifications: (i) to add new bound state(s), (ii)to remove bound state(s) and (iii) to leave the spectrum unaffected. To explain our findings with thehelp of an illustration, we have used point canonical transformation (PCT) to obtain the generalsolution of the position dependent mass Schrodinger equation corresponding to a potential andmass function. It is shown that our results are consistent with the formulation of type A N -foldsupersymmetry [14, 18] for the particular case N = 1 and N = 2 respectively. PACS numbers:
I. INTRODUCTION
There is a growing interest nowadays to design systems whose Hamiltonians have given spectral character-istics. In this context, the idea of designing potentials with prescribed energy spectra is worth investigating.Some progress in this area has been done by restricting the construction of potentials isospectral to a giveninitial one except a few energy values through the usage of Darboux transformation [1], factorization method[2], supersymmetric quantum mechanics (SUSYQM) [3, 4] and other related techniques. The underlying ideaof most of these procedures has been summarized in an algebraic scheme known as intertwining approach. Ingeneral, the objective of the intertwining is to construct the so-called intertwining operator L which performsan intertwining between an initial solvable Hamiltonian H and a new solvable one ¯ H with slightly modifiedspectrum such that LH = ¯ HL , ¯ ψ ( x ) = L ψ ( x ) (1)The ingredients to implement the intertwining are seed solutions of the initial stationary Schr¨odinger equationassociated to factorization energies less than or equal to the ground state energy of H . If L is a first orderdifferential operator, the standard SUSYQM, with supercharges built of first order Darboux transformationoperators, and the factorization method are recovered. On the other hand, if higher order differential operatorsare involved in the construction of L , it gives rise to higher order SUSYQM [24]. It is possible to generate ∗ Electronic address: bikash.midya @ gmail.com † Electronic address: barnana @ isical.ac.in ‡ Electronic address: raj @ isical.ac.in families of isospectral Hamiltonians by either of the two ways:(i) by iteration of first order Darboux transformations. Every chain of N first order Darboux transformationcreates a chain of exactly solvable Hamiltonians H , H , H ... H N [5]. Hence the intertwining operator L ( N ) between the initial Hamiltonian H and the final Hamiltonian H N can always be presented as a product of N first order Darboux transformation operators between every two juxtaposed Hamiltonians H , H , ..., H N , L ( N ) = L N L N − ... L L , H p L p = L p H p − , p = 1 , , ..., N (ii) by looking for the N -th order intertwining operator directly, expressing the intertwiner as a sum of N + 1terms g i ( x ) d i dx i , i = 0 , , · · · N , and solving the system of equations resulting from the intertwining relationshipfor the g i ( x )’s.At this point it is appropriate to mention that the quantum mechanical systems with position dependent(effective) mass [6] have attracted a lot of interest due to their relevance in describing the physics of manymicrostructures of current interest such as semiconductor heterostructures [7], quantum dots [8], helium clus-ters and metal crystals [9] etc. Recently, the intertwining operator method has been applied to Schr¨odingerequation with position dependent (effective) mass to construct first-order and chains(iterations) of first-orderDarboux transformations and the connection between the first-order Darboux transformation and effectivemass supersymmetry (factorization) was shown [10]. Subsequently Darboux transformations of arbitraryorder for position dependent mass Schr¨odinger equation was derived and factorization of the n-th order trans-formation into first order transformations and existence of a reality condition for the transformed potential wasshown [11]. In the standard supersymmetric (SUSY) approach for effective mass Hamiltonians [12], the ladderoperators are taken as first order differential operators similar to constant mass case, but now they depend onboth superpotential and mass function. As a result one obtains two partner potentials with the same effectivemass sharing identical spectra upto the zero mode of the supercharge. Second order supersymmetric approach(2-SUSY)was used in ref [13] for describing dynamics of a quantum particle with a position dependent mass.A compact expression for 2-SUSY isospectral pairs was derived in terms of senond order superpotential andthe mass function. A detailed analysis has been given about zero mode equations of second order superchargesand possible reduction of 2-SUSY scheme to first order SUSY. Recently, a generalization of standard SUSYknown as higher derivative SUSY or N -fold SUSY [14] was given for position dependent mass Hamiltonians.This method keeps the basic superalgebra intact but differs from the standard first order SUSY in that thesupercharges are represented as N -th order ( N >
1) differential operators.In this article an attempt is made to generate isospectral potentials of position dependent mass Schr¨odingerequation (PDMSE) by applying first and second order intertwining technique. Specifically, the second orderintertwiner is constructed by taking it as L = 1 M ( x ) d dx + η ( x ) ddx + γ ( x )where M ( x ) is the mass function and η ( x ), γ ( x ) are to be determined. Substituting this in the intertwiningrelationship (1), we have been able to solve the apparently intricate system of equations for η ( x ) and γ ( x )by assuming an ansatz. As mentioned earlier, the closed form formulas for N -th order intertwining operatorsand a pair of isospectral Hamiltonians with position dependent mass was already reported by Tanaka [14], foran arbitrary value of N without recourse to any ansatz.The motivation for constructing second order intertwining operator directly comes from the observationthat although an N -th order intertwining operator can be expressed formally as a product of N first orderintertwining operators, it does not necessarily mean that a system constructed by an N -th order intertwiningoperator is equivalent to one constructed by N successive applications of each 1st order operator. In fact, itwas shown in ref [15], by comparing the two approaches in the constant mass scenario, that the former is moregeneral than the latter. The advantage of the direct method used here over the iterative method is that onecan generate second-order isospectral partner potentials directly from the initial potential i.e. one need notgo through the first-order intertwining technique. We shall see that as in the case of constant mass scenario,it is possible to generate isospectral potentials with some spectral modification: (i) to add new bound state(s)(ii) to remove bound state(s) and (iii) to leave the spectrum unaffected. In this context a natural questionis: What is the utility of finding the isospectral potentials in the position dependent mass background? Toanswer this let us note that in different areas of possible applications of low dimensional structures as alreadymentioned, there is need to have energy spectrum which is predetermined. For example, in the quantum wellprofile optimization, isospectral potentials (by deleting or creating bound states at a particular energy of theoriginal potential) are generated through supersymmetric quantum mechanics. This is necessary because aparticular effect (such as intersubband optical transitions in a quantum well) may be grossly enhanced byachieving the resonance conditions e.g. appropriate spacings between the most relevant states and also bytailoring the wave functions so that the (combinations of) matrix elements relevant for this particular effectare maximized [16]. This is particularly important for studying higher order nonlinear processes.The organization of the paper is as follows: in section II and III we have explained the first and second-orderintertwining techniques respectively with possible spectral modifications, with the help of a suitable examplegiven in the Appendix. Also, the connection of our approach to type A N -fold SUSY is shown in these sections.Section IV is kept for discussions and comments. II. FIRST ORDER INTERTWINING
We consider the following two one-dimensional effective mass Schrodinger Hamiltonians (Bendaniel-Dukeform) [17] with the same spectrum but with different potential H ψ = Eψ , H = − (cid:20) ddx (cid:18) M ( x ) (cid:19) ddx (cid:21) + V ( x ) (2)and ¯ H ¯ ψ = E ¯ ψ , ¯ H = − (cid:20) ddx (cid:18) M ( x ) (cid:19) ddx (cid:21) + ¯ V ( x ) (3)We connect the Hamiltonians (2) and (3) by means of the intertwining technique. To this end, we look for anoperator L that satisfies the relation (1). Without loss of generality let us consider the first order intertwiningoperator [10], as L = 1 p M ( x ) ddx + A ( x ) (4)Now using the intertwining relation (1) and equating the coefficients of like order of derivatives we obtain¯ V = V + 2 A ′ √ M − M ′ M + M ′′ M (5)and A ( ¯ V − V ) = − A ′ M ′ M + V ′ √ M + A ′′ M (6)where ‘prime’ denotes differentiation with respect to x .Now using (5) , equation (6) reduces to A ′′ √ M − A ′ M ′ M − AM ′′ M + 3 AM ′ M − AA ′ + V ′ = 0 (7)Integrating equation (7) we get A ′ √ M − AM ′ M − A + V = µ (8)where µ is a constant of integration. Now we substitute A ( x ) = − K √ M (9)where K=K(x) being an auxiliary function, in equation (8) we obtain the following Riccati equation − K ′ M + KM ′ M − K M + V = µ (10)The equation (10) can be linearized by the substitution K ( x ) = U ′ ( x ) U ( x ) . Substituting this value of K in equations(9) and (10) we get A ( x ) = − U ′ √ M U (11)and − M U ′′ − (cid:18) M (cid:19) ′ U ′ + V U = µ U (12)respectively. The equation (12) is similar to equation (2) with E = µ , µ is sometimes called factorizationenergy and U ( x ) is called seed solution. It should be noted here that U ( x ) need not be normalizable solutionof (12). However, for A ( x ) to be well defined (without singularity), U must not have any zeroes on the realline. For this, we shall restrict µ ≤ E throughout this article, E being the ground state energy eigenvalues ofthe equation (2). Once we can determine the solution U of (12) then we shall able to construct the intertwiner L , the isospectral partner ¯ V and its bound state eigenvalues ¯ ψ ( x ) with the help of following relations¯ V = V − U ′′ M U + 2 U ′ M U + M ′ U ′ M U + M ′′ M − M ′ M (13) L = 1 √ M (cid:18) ddx − U ′ U (cid:19) (14)and ¯ ψ ( x ) ∝ L ψ = 1 √ M (cid:20) ddx − ( ln U ) ′ (cid:21) ψ (15)The intertwiner L cannot be used to generate wave function of ¯ H at the factorization energy µ , because LU = 0. We are showing below with the help of supersymmetry how ¯ ψ µ can be obtained from the relation L † ¯ ψ µ = 0, L † being the adjoint of L and is given by L † = 1 √ M (cid:18) − ddx − U ′ U + M ′ M (cid:19) (16)For this we calculate L † L and LL † given by L † L = − M d dx + M ′ M ddx + (cid:18) U ′′ M U − M ′ U ′ M U (cid:19) (17)and LL † = − M d dx + M ′ M ddx + U ′ M U − U ′′ M U + M ′′ M − M ′ M ! (18)Now from the equation (12) we have V = U ′′ M U − M ′ U ′ M U + µ (19)Substituting this value of V in (13) we obtain¯ V = − U ′′ M U + 2 U ′ M U + M ′′ M − M ′ M + µ (20)Now using (19) and (20) in (17) and (18) we get L † L = − M d dx + M ′ M ddx + V − µ = H − µ (21)and LL † = − M d dx + M ′ M ddx + ¯ V − µ = ¯ H − µ (22)respectively. It is clear from the equation (22) that the wave function of ¯ H at the factorization energy µ canbe obtained by L † ¯ ψ µ = 0 i.e., ¯ ψ µ ∝ exp (cid:20)Z (cid:18) − U ′ U + M ′ M (cid:19) dx (cid:21) = √ M U (23)It is to be noted that if U corresponds to the bound state of H , the wave function ¯ ψ µ ( x ) defined in (23)is notnormalized so that µ does not belong to the bound state spectrum of ¯ H . If U corresponds to the ground statewavefunction of H then then the potential ¯ V has no new singularity, except the singularity due to V , provided M is not singular and M = 0 . However, if we consider U to an arbitrary state other than ground state of H then ¯ V might contain extra singularities, which are not present in V . If U is nodeless and unbounded atthe both end points then ¯ ψ µ ( x ) defined in (23) is normalizable, so that µ can be included in the bound statespectrum of H to generate ¯ V .
In this case maximal set of bound state wavefunctions of ¯ H are given by { ¯ ψ µ , L ψ } . A. First-order intertwining and type A 1-fold SUSY
To show that the results obtained in the previous section are consistent with the results of type A N -foldSUSY, we are going to mention the brief results of type A N -fold SUSY formalism ( for details see [14] andreferences there). Type A N -fold SUSY is characterized by the type A monomial space¯ ν N = h , z, ...z N − i (24)preserved by ˜ H N : ˜ H −N = − A ( z ) d dz − B ( z ) ddz − C ( z ) (25)where A ( z ) = a z + a z + a z + a z + a , N ≥ B ( z ) = Q ( z ) − N − A ′ ( z ) C ( z ) = ( N −
N − A ′′ ( z ) − ( N − Q ′ ( z ) + RQ ( z ) = b z + b z + b , N ≥ R , a i , b i are being constants. Applying the algorithm for constructing type A N -fold SUSY in PDM system[14], one can construct the most general form of type A N -fold SUSY PDM quantum systems ( H , ¯ H , L ) orequivalently ( H + N , H −N , P N ) : H ±N = − M d dx + M ′ M ddx + V ± ( x ) (27) P N = M ( x ) − N N − Y k =0 (cid:20) ddx + W ( x ) − N M ′ ( x )4 M ( x ) + N − − k z ′′ ( x ) z ′ ( x ) (cid:21) (28)where V + = V − + 2 N (cid:18) W ′ ( x ) M ( x ) − M ′ ( x ) W ( x )2 M ( x ) (cid:19) (29) W ( x ) = d W −N ( x ) dx − z ′′ ( x ) z ′ ( x ) + M ′ ( x )2 M ( x ) (30) d W −N ( x ) dx = z ′′ ( x )2 z ′ ( x ) − M ( x ) B ( z )2 z ′ ( x ) − M ′ ( x )2 M ( x ) (31) z ′ ( x ) = M ( x ) A ( z ) (32)and the product of operators are ordered as N − Y k =0 F k = F N − F N − ...F The solution space of the type A Hamiltonians H ±N are given by ν ±N = e −W ±N h , z, ..., z N i| z = z ( x ) , where W + N = −W −N + ( N − ln | z ′ ( x ) | − N ln | M ( x ) | (33)It is easily seen that ( ¯ V − V ) obtained in (13) coincides with ( V + − V − ) in equation (29) (with N = 1) if onetakes (comparing L with P ) ddx ln U ( x ) = − W ( x ) + M ′ ( x )4 M ( x ) (34) B. Example of first-order intertwining
It may be emphasized that the results mentioned in section II are most general and valid for any potential V ( x ). However to illustrate the above procedure with the help of an example we shall need non-normalizablesolutions of (12) corresponding to a particular mass function M ( x ). In Appendix A we have used pointcanonical transformation approach(PCT) to solve the equation (12). Here we are going to construct theisospectral partners of the following potential obtained in Appendix A (we have considered p = λ = 1 forsimplicity) V ( x ) = [( a + b − c ) − e x + c ( c − e − x (35)corresponding to the mass function M ( x ) = 14 sech (cid:18) x (cid:19) (36)The bound state solutions and eigenstates of the equation (2) are given by (see Appendix A) ψ n ( x ) = (cid:18) (2 n + σ + δ + 1) n ! Γ( n + σ + δ + 1)Γ( n + σ + 1)Γ( n + δ + 1) (cid:19) / e ( σ +1)2 x (1 + e x ) σ + δ +22 P ( σ,δ ) n (cid:18) − e x e x (cid:19) (37)and E n = n + n ( σ + δ + 1) + ( σ + 1)( δ + 1)2 , n = 0 , , , ... (38)respectively, where b = 1 − a + σ + δ, c = 1 + σ with c > and a + b − c + > . The seed solution U ( x ) andfactorization energy µ are given by U ( x ) = α e c x (1 + e x ) a + b +12 F (cid:18) a, b, c, e x e x (cid:19) + β e ( − c ) x (1 + e x ) a + b − c +32 F (cid:18) a − c + 1 , b − c + 1 , − c, e x e x (cid:19) (39) µ = − ab + ( a + b + 1) c − c U ( x ) given in (39), at both end points ±∞ are given by[19] U ( x ) ∼ ( A α + B β ) e − a + b − c +12 x + ( A α + B β ) e − c − a − b +12 x as x → ∞ (41)where A = Γ( c )Γ( c − a − b )Γ( c − b )Γ( c − a ) , B = Γ(2 − c )Γ( c − a − b )Γ(1 − a )Γ(1 − b ) A = Γ( c )Γ( a + b − c )Γ( a )Γ( b ) , B = Γ(2 − c )Γ( a + b − c )Γ( a − c + 1)Γ( b − c + 1)and U ( x ) ∼ αe c x + β e (1 − c ) x as x → −∞ (42)From these asymptotic behaviors it is clear that U ( x ) will unbounded at x → ∞ if | a + b − c | > x → −∞ if c < >
2. Therefore U ( x ) will nodeless at the finite part of the x axis if A α + B β , A α + B β , α and β are all positive and | a + b − c | > c < > Deletion of the initial ground state :
In this case the factorization energy µ is equal to the ground stateenergy E giving a and / or b = 0 and U ( x ) becomes the ground state wavefunction ψ ( x ) which is obtainedfrom (37) as U ( x ) = ψ ( x ) ∝ e c x (1 + e x ) a + b +12 (43)The isospectral partner of V ( x ) given in equation (35), is obtained using equation (13), (35), (36) and (43)and is given by ¯ V ( x ) = c − e − x + ( a + b − c )(2 + a + b − c )4 e x + a + b a → a + 1 , b → b + 1 , c → c + 1, this property is known as shape invariance [4]. Since ψ ( x ) is bounded solution¯ ψ µ ( x ) = √ Mψ is unbounded at x → ±∞ , so we have deleted the ground state energy of H to obtain ¯ V ( x ).Therefore the eigenvalues of ¯ H are given by¯ E n = E n +1 = ( n + 1) + ( n + 1)( a + b ) + c ( a + b − c + 1)2 , n = 0 , , ... (45)Corresponding bound state wavefunctions of ¯ V ( x ) are obtained using equation (15) as¯ ψ n ( x ) ∝ e (1+ c ) x (1 + e x ) ( a + b +52 ) sech ( x ) P ( c,a + b − c +1) n (cid:16) − tanh x (cid:17) , n = 0 , , ... (46)We have plotted the potentials V ( x ) given in (35) and ¯ V ( x ) given in (44) for a = 5 , b = 0 , c = 3 , α = 1 , β = 0in figure 1. E = X - - - FIG. 1: Plot of the potential V ( x ) (solid line) given in (35) and its first order isospectral partners ¯ V ( x ) (dashed line)given in (44) by deleting the ground state E = 4 .
5, we have considered here a = 5 , b = 0 , c = 3 , α = 1 , β = 0 . Strictly isospectral potentials :
The strictly (strict in the sense that the spectrum of the initial potentialand its isospectral potential are exactly the same) isospectral potentials can be generated with the help ofthose seed solutions which vanish at one of the ends of the x -domain. Now for β = 0 and α >
0, it is seenfrom (41) that U is unbounded at x → ∞ if | a + b − c | >
1. But the solution (37) become unbounded for a + b − c < −
1. So we must take a + b − c > . On the other hand from (42) it is observed that U ( x ) → x → −∞ if c < c > ψ n ( x ) are not normalizable for the values of c < c >
0. So U ( x ) vanishes at x → −∞ and unbounded at x → ∞ if a + b − c > c > . In this case the spectrum ofthe isospectral potential as well as original potential are identical i.e. E n = ¯ E n , n = 0 , , ... Considering theseed solution as U ( x ) = e c x (1 + e x ) a + b +12 F (cid:18) a, b, c, e x e x (cid:19) , a + b − c > , c > V ( x ) = 18 [2 c ( c − e − x + 2(( a + b − c ) − e x − sech ( x ) c (1+ c ) F ( a,b,c, ex ex ) {− a b (1 + c ) (cid:16) F (cid:16) a, b, c, e x e x (cid:17)(cid:17) + 4 abc F (cid:16) a, b, c, e x e x (cid:17)(cid:18) ( a + 1)( b + 1) (cid:16) F (cid:16) a, b, c, e x e x (cid:17)(cid:17) − (1 + c ) sinhx (cid:16) F (cid:16) a + 1 , b + 1 , c + 1 , e x e x (cid:17)(cid:17) (cid:19) − c (1 + c ) cosh x (cid:16) F (cid:16) a, b, c, e x e x (cid:17)(cid:17) (cid:0) ( a + b ) cosh ( x ) + (1 + a + b − c ) sinh ( x ) (cid:1) } ] (47)In particular for a = 3 , b = 5 , c = 4 , α = 1 , β = 0 and using (35), (13) we have obtained V ( x ) = 14 (23 coshx + 7 sinhx )¯ V ( x ) = e − x + 2 e x + e x − e x (4+3 e x ) (48)respectively, which are plotted in figure 2. In this case eigenfunctions and eigenvalues of the above partnerpotential ¯ V ( x ) are given by¯ ψ n ( x ) ∝ e x (cid:16) (2 + e x )( n + 7) P (4 , n − (cid:0) − tanh (cid:0) x (cid:1)(cid:1) + (1 + e x )(5 + 3 e x ) P (3 , n ( − tanh ( x )) (cid:17) (1 + e x ) (2 + e x ) sech (cid:0) x (cid:1) ¯ E n = n + 8 n + 10 , n = 0 , , ... (49)respectively. Creation of a new ground state :
In this case we shall consider µ < E . The new state can be createdbelow the ground state of the initial potential with the help of those seed solutions which satisfies the followingtwo conditions: (i) it should be nodeless throughout the x -domain and (ii) it should be unbounded at boththe end points of the domain of definition of the given potential V ( x ). From the asymptotic behaviors ofthe seed solution U , given in equations (41) and (42) we have, for | a + b − c | > c < c >
2, the above two conditions are satisfied. But to get ψ n ( x ) as physically acceptable, we shall take c > a + b − c > . In this case the spectrum of the partner potential is { µ, E n , n = 0 , , ... } , E n beingthe energy eigenvalues of the original potential V ( x ) given in (35). Corresponding bound state wavefunctionsare { ¯ ψ µ ( x ) , ¯ ψ n ( x ) , n = 0 , , , ... } , where ¯ ψ µ and ¯ ψ n are given by (23) and (15) respectively.For a + b − c > , c > U given in (39), the general expression of the isospectral potentialbecomes too involved so instead of giving the explicit expression of the partner potential we have plotted infigure 3 the original potential V ( x ) given in (35) and its partner potential ¯ V ( x ) (which is obtained using (13))0 X - - FIG. 2: Plot of the potential V ( x ) (solid line) and its first order isospectral partner (dashed line) given in (48). considering the particular values a = 2 . , b = 20 , c = 4 . α = β = 1. In this case the energy eigenvaluesof ¯ V ( x ) are given by¯ E n = {− . , E n , n = 0 , , ... } = {− . , n + 22 . n + 42 . , n = 0 , , ... } (50)Corresponding eigenfunctions can be obtained using the formulae (23) and (15). Μ = - X - - - - - FIG. 3: Plot of the potential V ( x ) (solid line) given in (35) and its first order isospectral partner ¯ V (dashed line) byinserting the state µ = − .
32. We have considered here a = 2 . , b = 20 , c = 4 . , α = β = 1. III. SECOND ORDER INTERTWINING
Now we assume the existence of a second order intertwining operator L = 1 M d dx + η ( x ) ddx + γ ( x ) (51)where η ( x ) , γ ( x ) are to be determined. Substitution of this intertwiner in equation (1) and comparison of thecoefficients of like order derivatives leads to a set of following equations¯ V = V + 2 η ′ + M ′ M η − M ′ M + 2 M ′′ M (52)( ¯ V − V ) η = 2 V ′ M + 2 γ ′ M + η ′′ M − η ′ M ′ M + M ′′ ηM − M ′ ηM + 6 M ′ M − M ′ M ′′ M + M ′′′ M (53)( ¯ V − V ) γ = V ′′ M + V ′ η + γ ′′ M − M ′ γ ′ M (54)Now using (52) the equations (53) and (54) reads2 ηη ′ + M ′ η M − ηM ′ M + 2 M ′′ ηM − γ ′ M + M ′ η ′ M + 2 M ′ ηM − η ′′ M − M ′′ ηM − V ′ M − M ′ M + 6 M ′ M ′′ M − M ′′′ M = 0 (55)and γ (cid:18) η ′ + M ′ ηM − M ′ M + 2 M ′′ M (cid:19) + M ′ γ ′ M − γ ′′ M − ηV ′ − V ′′ M = 0 (56)respectively. Equation (55) can be integrated to obtain γ = M η M ′ η M − η ′ − V + M ′ M − M ′′ M + C (57)where C is an arbitrary constant. Using (57) in (56) we obtain η ′′′ M + M η ′ η − ηη ′ M ′ M − η ′ − η ′ V + 5 η ′ M ′ M − η ′ M ′′ M + η M ′ − η M ′ M − ηV M ′ M − ηM ′ M + η M ′′ M + 5 ηM ′ M ′′ M − η ′′ M ′ M − ηη ′′ − ηV ′ − ηM ′′′ M + 2 C η ′ + C ηM ′ M + 3 V M ′ M − M ′ M + 49 M ′ M ′′ M − C M ′ M − V M ′′ M − M ′′ M + 2 C M ′′ M − V ′ M ′ M − M ′ M ′′′ M + M ′′′′ M = 0 (58)Multiplying by (cid:16) ηM + M ′ M (cid:17) , above equation (58) can be integrated to obtain ηη ′′ − η ′ − η ′ η M + η M − M η V + C M η + C M ′ M + 2 C M ′ ηM − M ′ V ηM − M ′ VM − M ′′ η M + M ′ η M ′ ηm − M ′ ηη ′ M + 5 M ′ η M + M ′ η ′′ M − M ′′ η ′ M + M ′′′ η M − M ′ M ′′ ηM + 3 M ′ M − M ′ M ′′ M − M ′′ M + M ′ M ′′′ M + C = 0 (59)2where C is the constant of integration. For a given potential V ( x ), the new potential ¯ V ( x ) and γ ( x ) can beobtained from (52) and (57) if the solution η ( x ) of (59) is known. To obtain η ( x ) we take the Ans¨atz η ′ = M η + 2 (cid:18) η + M ′ M (cid:19) τ + M ′ M η + 2 M ′ M − M ′′ M + ξ (60)where ξ is a constant to be determined and τ is a function of x . Using above ans¨atz in equation (59) we obtainthe following equation M (cid:18) τ ′ M + τ M − M ′ τM − V + C − ξ (cid:19) η + 2 M ′ M (cid:18) τ ′ M + τ M − M ′ τM − V + C − ξ (cid:19) η + M ′ M (cid:18) τ ′ M + τ M − M ′ τM − V + C − ξ (cid:19) + (cid:18) C − ξ (cid:19) = 0 (61)Since equation (61) is valid for arbitrary η , the coefficients of each power of η must vanish, which give ξ = 4 C and τ ′ M + τ M − M ′ τM − V + C − ξ µ = C − ξ , the above equation can be written as τ ′ M + τ M − M ′ τM = V − µ , µ = C − ξ C ∓ p C (63)The equation (63) is a Riccati equation which can be linearized by defining τ = U ′ U . Making this change inequation (63) we obtain − M U ′′ − (cid:18) M (cid:19) ′ U ′ + V U = µ U (64)Depending on whether C is zero or not, ξ vanishes or takes two different values ±√ C . If C = 0, we needto solve one equation of the form (63) and then the equation (60) for η ( x ). If C = 0, there will be twodifferent equations of type (63) for two factorization energies µ , = C ∓ √ C . Once we solve them, it ispossible to construct algebraically a common solution η ( x ) of the corresponding pair of equations (60). Thereis an obvious difference between the real case with C > C <
0; thus there follows anatural scheme of classification for the solutions η ( x ) based on the sign of C . In our present article we shallnot discuss the case C = 0 . (i) Real Case ( C > µ , ∈ R , µ = µ . Let the corresponding solutions of the Riccati equation (63) be denoted by τ , ( x ) . Now the associated pair of equations (60) become η ′ = M η + 2 (cid:18) η + M ′ M (cid:19) τ + M ′ M η + 2 M ′ M − M ′′ M + µ − µ (65)and η ′ = M η + 2 (cid:18) η + M ′ M (cid:19) τ + M ′ M η + 2 M ′ M − M ′′ M + µ − µ (66)3respectively. Subtracting (65) from (66) and using (64) we obtain η ( x ) as η ( x ) = µ − µ τ − τ − M ′ M = − W ′ ( U , U ) M W ( U , U ) (67)where U , U are the seed solutions of the equation (64) corresponding to the factorization energy µ and µ respectively and W ( U , U ) = U U ′ − U ′ U , is the Wronskian of U and U .Now it is clear from (52) and (67) that mass function M ( x ) is nonsingular and does not vanish at the finitepart of the x -domain, so that the new potential ¯ V ( x ) has no extra singularities (i.e. the number of singularitiesin V and ¯ V remains the same) if W ( U , U ) is nodeless there. The spectrum of ¯ H depends on whether or notits two eigenfunctions ¯ ψ µ , which belongs as well to the kernel of L † can be normalized [21], namely L † ¯ ψ µ j = 0 and ¯ H ¯ ψ µ j = µ j ¯ ψ µ j , j = 1 , L † is the adjoint of L and is given by [20] L † = 1 M d dx − (cid:18) η + 2 M ′ M (cid:19) ddx + (cid:18) M ′ M − M ′′ M − η ′ + γ (cid:19) For j = 1 the explicit expression of the two equation mentioned in (68) are1 M d ψ µ dx − (cid:18) η + 2 M ′ M (cid:19) dψ µ dx + (cid:18) M ′ M − M ′′ M − η ′ + γ (cid:19) ψ µ = 0 (69)and − M ψ ′′ µ − (cid:18) M (cid:19) ′ ψ ′ µ + ( ¯ V − µ ) ψ µ = 0 (70)respectively. Adding (69) from (70) we obtain − (cid:18) M ′ M + η (cid:19) dψ µ dx + (cid:18) ¯ V − µ + 2 M ′ M − η ′ + γ − M ′′ M (cid:19) ψ µ = 0 (71)Substituting the values of ¯ V and γ from (52) and (57) with 2 C = µ + µ , in the above equation (71), we get ddx ( logψ µ ) = η ′ + 3 η M ′ M + M ′′ M + M η + 2( C − µ )2( η + M ′ M ) (72)Now using our ans¨atz (60) in (72) and then integrating we obtain ψ µ ∝ M (cid:16) η + M ′ M (cid:17) U ∝ M U W ( U , U ) (73)Above procedure can be applied to obtain ¯ ψ µ as¯ ψ µ ∝ ηM + M ′ M U ∝ M U W ( U , U ) (74)If both ¯ ψ µ , are normalizable then we get the maximal set of eigenfunctions of ¯ H as { ¯ ψ µ , ¯ ψ µ , ¯ ψ n ∝ L ψ n } .Among the several spectral modifications which can be achieved through the real second order SUSYQM for4PDMSE, some cases are worth to be mentioned. Deletion of first two energy levels :
For µ = E and µ = E the two solutions of equation(64) are the normalizable solutions of equation (2) i.e, U = ψ ( x ) and U = ψ ( x ) respectively. Itturns out that the Wronskian is nodeless but two solutions ¯ ψ µ and ¯ ψ µ are non-normalizable. Thus Sp ( ¯ H ) = Sp ( H ) − { E , E } = { E , E , E , ... } , i.e., the two levels E and E are deleted to generate ¯ V .
Isospectral transformations :
If we take µ < µ < E and choose U and U such way that either U , ( x l ) = 0 or U , ( x r ) = 0, x l and x r being the end points of the domain of definition of V ( x ), thenthe Wronskian W ( U , U ) vanishes at x l or x r . Hence ¯ ψ µ and ¯ ψ µ become non-normalizable so that Sp ( H ) = Sp ( ¯ H ). Creation of two new levels below the ground state :
For µ < µ < E and choosing U and U in such way that U has exactly one node and U is nodeless then the Wronskian W ( U , U ) becomesnodeless, also two wavefunctions ¯ ψ µ and ¯ ψ µ are normalizable. Therefore the spectrum of ¯ H becomes Sp ( ¯ H ) = Sp ( H ) S { µ , µ } = { µ , µ , E n , n = 0 , , ... } i.e. two new levels have been inserted to the spectrumof V ( x ) to obtain ¯ V ( x ) . (ii) Complex case ( C < C < µ and µ become complex. In order to construct real ¯ V we shallchoose µ and µ as complex conjugate to each other i.e, µ = µ ∈ C and µ = ¯ µ . For the same reason weshall take τ ( x ) = τ ( x ) and τ ( x ) = ¯ τ ( x ) . Hence the real solution η ( x ) of (60) generated from the complex τ ( x ) of (63) becomes η ( x ) = µ − ¯ µτ − ¯ τ − M ′ M = Im ( µ ) Im ( τ ) − M ′ M = − W ′ ( U , ¯ U ) M W ( U , ¯ U ) (75)Defining w ( x ) = W ( U , ¯ U ) M ( µ − ¯ µ ) , η ( x ) becomes η ( x ) = − w ′ M w − M ′ M (76)For the factorization energies µ and ¯ µ the equation (64) becomes − M U ′′ − (cid:18) M (cid:19) ′ U ′ + V U = µ U and − M ¯ U ′′ − (cid:18) M (cid:19) ′ ¯ U ′ + V ¯ U = ¯ µ ¯ U Multiplying first equation by ¯ U and second equation by U and then subtracting we obtain W ′ ( U , ¯ U ) M ( µ − ¯ µ ) − M ′ W ( U , ¯ U ) M ( µ − ¯ µ ) = |U| (77)Using above relation (77) we have w ′ ( x ) = W ( U , ¯ U ) M ( µ − ¯ µ ) − M ′ W ( U , ¯ U ) M ( µ − ¯ µ ) = |U| (78)which implies that w ( x ) is a non-decreasing function. So it is sufficient to chooselim x → x l U = 0 or lim x → x r U = 0 (79)for the Wronskian W to be nodeless. It is to be noted here that in this case we can only construct potentialswhich are strictly isospectral with the initial potential.5 A. Second-order intertwining and type A 2-fold SUSY
The second-order intertwiner L in equation (51) coincides with P given in equation (28) if one takes η ( x ) = 2 W ( x ) M ( x ) − M ′ ( x ) M ( x ) (80)It is now easy to verify that for this η ( x ), ( V + − V − ) given in (29) (with N = 2) agree with ( ¯ V − V ) given in(52).Now it is to be shown that the Hamiltonian H given in equation (2) admits two eigenfunctions U , ( x )corresponding to two factorization energies µ , respectively i.e., H U i ( x ) = µ i U i ( x ) , i = 1 , z ( x ) = U ( x ) U ( x ) , W − ( z ) ≡ W ( z ) = − ln U ( x ) (82)For this W ( z ), it is evident that the gauged Hamiltonian ˜ H − defined by˜ H − = e W H e −W (83)must be diagonal in the basis ˜ ν = h , z i because of the assumption (81) and the choice (82). From equation(31), its immediate consequence (for N = 2) is B ( z ) = z ′′ ( x ) M ( x ) − z ′ ( x ) M ′ ( x ) M ( x ) − z ′ ( x ) M ( x ) d W ( z ) dz (84)Using equations (82) and (81) in the above equation (84) it can be shown that B ( z ) = ( µ − µ ) z ( x ) (85)For this value of B ( z ) it is also easy to verify that the expression of η ( x ) in equation (80) and (67) are same.Now it is evident that the gauged Hamiltonian ˜ H − preserves the vector space ˜ ν = h , z i . Hence it is possibleto get type A 2-fold SUSY system ( H , ¯ H , L ) following the prescription given in ref. [14], with the choice of z ( x ), W ( z ) and ˜ H − given by (82) and (25) respectively. B. Example of second-order intertwining for real factorization energies
It may be emphasized that the results mentioned in section III are most general and valid for any potential V ( x ). However to illustrate the above procedure with the help of an example we shall need non-normalizablesolutions of (64) (which is similar to equation (12) but with two factorization energies) corresponding to aparticular mass function M ( x ). To illustrate the second order intertwining with an example we have consideredthe potential (35) as an initial potential. Corresponding seed solution for the factorization energy µ = µ whichis obtained in Appendix A, is µ = − ab + ( a + b + 1) c − c U ( x ) = α e c x (1 + e x ) a + b +12 F (cid:18) a, b, c, e x e x (cid:19) + β e ( − c ) x (1 + e x ) a + b − c +32 F (cid:18) a − c + 1 , b − c + 1 , − c, e x e x (cid:19) (87)We notice that the potential (35) and corresponding Hamiltonian are invariant under the transformation a → a + ν and b → b − ν , ν ∈ R − { } . But the solution (39) of the corresponding Schr¨odinger equationchanges to U ( x ) = α e c x (1 + e x ) a + b +12 F (cid:18) a + ν, b − ν, c, e x e x (cid:19) + β e ( − c ) x (1 + e x ) a + b − c +32 F (cid:18) a + ν − c + 1 , b − ν − c + 1 , − c, e x e x (cid:19) (88)and the corresponding factorization energy is given by µ = − ab + ( a + b + 1) c − c ν ( a − b ) + ν (89)Thus the general solutions of the equation (64) for the two factorization energies µ and µ , are given by (87)and (88)respectively. The asymptotic behaviors of the seed solution U remains same as U , which are givenin (41) and (42). Deletion of first two energy levels :
Let us take µ = E and µ = E , U = ψ ( x ) and U = ψ ( x ) whichare given in (37). The Wronskian W ( U , U ) is given by W ( U , U ) ∝ e ( c +1) x (1 + e x ) a + b +3 (90)which is nodeless and bounded in ( −∞ , ∞ ) as c > − and a + b − c + > V ( x ) is obtained using equation (52) andis given by ¯ V ( x ) = 14 (cid:2)(cid:0) c − c ( a + b + 2) + ( a + b + 1)( a + b + 3) (cid:1) e x + c ( c + 2) e − x + 4( a + b + 1) (cid:3) (91)Clearly the eigenfunctions ¯ ψ µ ∝ M U W ( U , U ) and ¯ ψ µ ∝ M U W ( U , U ) of ¯ H associated to µ = E and µ = E arenot normalizable since lim x →−∞ , ∞ ¯ ψ µ , ( x ) = ∞ Thus Sp ( ¯ H ) = Sp ( H ) − { E , E } = { E , E ... } . In particular taking a = 5 , b = 0 , c = 3 we have plotted the potential V ( x ) and its second-order SUSY partner¯ V ( x ) given in (35) and (91) respectively, in figure 4. Strictly isospectral potentials :
The strictly isospectral partner potentials can be constructed by creatingtwo new energy levels in the limit when each seed vanishes at one of the ends of the x -domain. Now fromthe asymptotic behaviors of the seed solutions, we note that both the seed solutions vanish at x → −∞ for β = 0 , α > a + b − c > c >
0. Considering β = 0 , α = 1 in (87) and (88) we take two seed solution as U ( x ) = e c x (1 + e x ) a + b +12 F (cid:18) a, b, c, e x e x (cid:19) (92)and U ( x ) = e c x (1 + e x ) a + b +12 F (cid:18) a + ν, b − ν, c, e x e x (cid:19) (93)7 E = E = X - - FIG. 4: Plot of the original potential (solid line) for a = 5 , b = 0 , c = 3 and its first-order SUSY partner (dashed line)by deleting the ground state E = 4 . E = 4 . , E = 10 . . Since U , ( x ) → x → −∞ , from the expressions (73) and (74) we can conclude thatlim x →−∞ ¯ ψ µ , ( x ) = ∞ which implies that µ , does not belongs to Sp ( ¯ H ) i.e. ¯ V ( x ) is strictly isospectral to V ( x ) . Here the generalexpression of the partner potential is too involved so instead of giving the explicit expression we have consideredparticular values a = 3 , b = 5 , c = 4 , ν = 1 , α = 1 , β = 0. Corresponding expression of the partner potentialand its energy spectrum are¯ V ( x ) = 1 + 34 (9 coshx − sinhx ) , E n = ¯ E n = n + 8 n + 10 , n = 0 , , ... respectively. In figure 5, we have plotted the initial potential, its first and second-order strictly isospectralpartner potentials for the parameter values a = 3 , b = 5 , c = 4 , α = 1 , β = 0 , ν = 1. X - - FIG. 5: Plot of the original potential (solid line) and its first-order (dashed line) and second-order (dotted line) SUSYpartner by making the isospectral transformation for a = 3 , b = 5 , c = 4 , α = 1 , β = 0 , ν = 1. Creation of two new levels below the ground state :
Two energy levels can be created taking µ < µ < E and using those seed solutions U and U for which the Wronskian become nodeless. In this casethe expressions of the Wronskian contains several Hypergeometric function, so it is very difficult to mentionthe range of a, b, c and ν for which it is nodeless. In particular for a = 2 . , b = 20 , c = 4 . , α = 1 , β = 1 wehave the Wronskian is found to be nodeless. For the same values of a, b, c we have plotted the potential andits second order partner in figure 6. The second-order isospectral partner is obtained using equation (52). Μ = - Μ = - X - - - - - FIG. 6: Plot of the original potential (solid line) for a = 2 . , b = 20 , c = 4 . µ = − .
32 and second-order SUSY partner (dotted line) by creating two new levels µ = − . , µ = − . C. Example of second order intertwining for complex factorization energies
As mentioned earlier, in this case we can only construct the strictly isospectral partner potentials. Thecomplex factorization energy µ and µ given by equation (86) and (89), can be made conjugate to each otherin several ways. One of the way is by making following restrictions on a, b, c, ν : c ∈ R , Im ( a ) = − Im ( b ) , ν = Re ( b ) − Re ( a ). But in order to keep the initial potential real we have to made two more restrictions e.g. Re ( a ) + Re ( b ) − c > c > . In particular taking a = 6 . − i, b = 8 + 5 i, c = 4 . , ν = 1 . µ (= µ ) = − .
25 + 9 . i and µ (= ¯ µ ) = − . − . i. For these values of a, b, c, ν and α = 1 , β = 0 the seed solution U becomes U ( x ) = e . x (1 + e x ) .
55 2 F (cid:18) . − i, i, . , e x e x (cid:19) (94)Clearly U ( −∞ ) = 0 and |U| → ∞ as x → ∞ so this seed U and its conjugate ¯ U can be used to obtain thesecond-order SUSY partner potential ¯ V ( x ) with the help of equations (52) and (75). In figure 7 we have plottedthe initial potential V ( x ) given in (35) and its isospectral partner ¯ V ( x ) for the parameter values mentionedearlier. X - - FIG. 7: Plot of the original potential (solid line) for a = 6 . − i, b = 8 + 5 i, c = 4 . , ν = 1 . , µ = − .
25 + 9 . i, µ = − . − . i. and its second-order isospectral partner (dashed line). IV. SUMMARY AND OUTLOOK
In this article we have discussed the possibilities for designing quantum spectra of position dependentmass Hamiltonians offered by the intertwining technique. For doing this, we start with the non-normalizablesolution of position dependent mass Schr¨odinger equation with the initial potential (obtained by using thepoint canonical transformation approach). To generate spectral modifications by first order intertwining, wehave used solutions to the position dependent mass Schr¨odinger equation corresponding to factorization energy(not belonging to the physical spectrum of the initial problem) less than or equal the ground state energyin order to avoid singularity in the isospectral partner potential provided the mass function is not singularand is not equal to zero in the real line. Thus it is possible to generate isospectral partner potentials (a)with the ground state of the original potential deleted (b) with a new state created below the ground stateof the original potential (c) with the spectrum of the original potential unaffected. In ref [10], the first orderintertwining technique was illustrated by considering the free particle case.0In the case of second order intertwining, instead of using the iterative method used in [10], the secondorder intertwiner is constructed directly by taking it as second order linear differential operator with unknowncoefficients which are functions of x . The main advantage of this construction is that one can generate second-order isospectral partner potentials directly from the initial potential without generating first-order partnerpotentials. The apparently intricate system of equations arising from the intertwining relationship is solved forthe coefficients by taking an ansatz. In this case the spectral modifications are done by taking appropriatelychosen factorization energies which may be real or complex. For real unequal factorization energies, it ispossible to generate potentials (a) with deletion of first two energy levels (b) with two new levels embeddedbelow the ground state of the original potential (c) with identical spectrum as of the original potential. Forcomplex factorization energies, it is shown how to obtain strictly isospectral potentials. It must be mentionedhere that in all the above cases the conditions for having spectral modifications remain the same as in thecase of constant mass scenario [21] provided the mass function M is nonsingular and is not equal to zero inthe finite part of the real line.In this article, the equivalence of our formalism to type A N -fold ( N = 1 ,
2) SUSY in PDM backgroundis shown. Also, it is shown that an arbitrary one body quantum PDM Hamiltonian which admits two eigen-functions in closed form belongs to type A 2-fold SUSY as was previously done in constant mass scenario[18].Some of the interesting issues to be investigated in future are(i) to obtain spectral changes that appear above the ground state energy of the initial potential. Specifically,how to create/delete a pair of levels between any two neighboring initial ones, how to move an arbitrary levelor delete an arbitrary level. Specially interesting will be the possibility of embedding a single level at anyarbitrary position.(ii) to obtain spectral modifications when the two factorization energies are equal.
Appendix A: Construction of exactly solvable effective potential via PCT
In order to find the general (unbounded) solution of the equation (12) we shall use PCT method in PDMbackground [22] to solve this equations. Let us find the solution of equation (12) of the form U ( x ) = f ( x ) F ( a, b, c, g ) (A1)where f ( x ) , g ( x ) are two function of x to be determined and F ( a, b, c, g ) is the Hypergeometric function whichsatisfies second order differential equation of the type d Fdg + Q ( g ) dFdg + R ( g ) F = 0 , with Q ( g ) = c − ( a + b + 1) gg (1 − g ) and R ( g ) = − abg (1 − g ) (A2)Substituting equation (A1) in (12) we obtain d Fdg + (cid:18) g ′′ g ′ + 2 f ′ f g ′ − M ′ M g ′ (cid:19) dFdg + (cid:18) f ′′ f g ′ + ( µ − V ) Mg ′ − M ′ f ′ M f g ′ (cid:19) F = 0 (A3)Comparing equation (A2) and (A3) we get Q ( g ( x )) = g ′′ g ′ + f ′ fg ′ − M ′ Mg ′ R ( g ( x )) = f ′′ fg ′ + ( µ − V ) Mg ′ − M ′ f ′ Mfg ′ (A4)After simplification of the above equation (A4) we obtain f ( x ) ∝ s Mg ′ exp Z g ( x ) Q ( t ) dt ! (A5)1and µ − V = g ′′′ M g ′ − M (cid:18) g ′′ g ′ (cid:19) + g ′ M (cid:18) R − dQdg − Q (cid:19) − M ′′ M + 3 M ′ M (A6)respectively. Now in PCT approach there are many options for choosing M ( x ) [22], for example M ( x ) = λg ′ ( x ) , M = λg ′ ( x ), M = λg ′ ( x ) , λ being a constant. Here we choose M ( x ) = λg ′ ( x ). For this choice of themass function and using the values of Q ( g ) , R ( g ) given in equation (A2), equation (A6) reduces to µ − V = g ′ λ " − abg (1 − g ) − ( c − ( a + b + 1) g ) g (1 − g ) + a + b + 12 g (1 − g ) + c − ( a + b + 1) g g (1 − g ) − c − ( a + b + 1) gg (1 − g ) (A7)Now in order to generate a constant term on the right hand side of the above equation which will correspondto µ on the left-hand side, we set g ′ λg (1 − g ) = p , where p is a positive constant. This gives g ( x ) = e pλx e pλx and M ( x ) = pλ sech (cid:18) pλ x (cid:19) , − ∞ < x < ∞ (A8)For these values of g ( x ) and M ( x ) we obtain from equation (A7) new potential V ( x ) and factorization energy µ as V ( x ) = [( a + b − c ) − p e pλx + cp ( c − e − pλx , − ∞ < x < ∞ (A9)and µ = − abp + ( a + b + 1) cp − c p f ( x ) = e cpλ x (1 + e pλx ) a + b +12 Hence the solution of the equation (12) at the factorization energy µ is given by U ( x ) = e cpλ x (1 + e pλx ) a + b +12 F (cid:18) a, b, c, e pλx e pλx (cid:19) (A11)Another linearly independent solution of (12) at the same factorization energy can be written as [19] U ( x ) = e pλ ( − c ) x (1 + e pλx ) a + b − c +32 F (cid:18) a − c + 1 , b − c + 1 , − c, e pλx e pλx (cid:19) The linear combination of above two solutions can be taken as the most general non-normalizable solution ofthe equation (12) at the factorization energy µ , and is U ( x ) = α e cpλ x (1 + e pλx ) a + b +12 F (cid:18) a, b, c, e pλx e pλx (cid:19) + β e pλ ( − c ) x (1 + e pλx ) a + b − c +32 F (cid:18) a − c + 1 , b − c + 1 , − c, e pλx e pλx (cid:19) (A12)2where α and β are two arbitrary constants. Consequently the bound state solutions of the equation (2) for thepotential (A9), are obtained from equation (A12) by putting α = 1 , β = 0 and a = − n, b = 1 − a + σ + δ, c =1 + σ (see 15.4.6 of ref. [19]) ψ n ( x ) = (cid:18) pλ (2 n + σ + δ + 1) n ! Γ( n + σ + δ + 1)Γ( n + σ + 1)Γ( n + δ + 1) (cid:19) / e pλ ( σ +1)2 x (1 + e pλx ) σ + δ +22 P ( σ,δ ) n (cid:18) − e pλx e pλx (cid:19) (A13)and the energy eigenvalues are given by E n = n p + np ( σ + δ + 1) + ( σ + 1) p ( δ + 1)2 , n = 0 , , , ... (A14)It should be mentioned here that for ψ n ( x ) to be a physically acceptable solution it should satisfy the followingtwo conditions:(i) It should be square integrable over domain of definition D of M ( x ) and ψ ( x ) i.e., Z D | ψ n ( x ) | dx < ∞ (ii) The Hermiticity of the Hamiltonian (2) in the Hilbert space spanned by the eigenfunctions of the potential V ( x ) is ensured by the following extra condition [23] | ψ n ( x ) | p M ( x ) → V ( x ) and ψ n ( x ) are defined. This condition imposes an additionalrestriction whenever the mass function M ( x ) vanishes at any one or both the end points of D . In orderto satisfy this two conditions we have to impose a restriction σ > − and δ > − or equivalently c > and a + b − c + > . Acknowledgments
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