ASYMPTOTIC ITERATION METHOD APPLIED TO BOUND-STATE PROBLEMS WITH UNBROKEN AND BROKEN SUPERSYMMETRY 1 2 ¨ ´ O. OZER , G. LEVAI 1
Department of Engineering Physics, Engineering Faculty, University of Gaziantep, 27310 Gaziantep, Turkey Email:
[email protected] 2 Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI), PO Box 51, H-4001 Debrecen, Hungary Email:
[email protected] Received February 28, 2011
The bound-state spectra for some potentials with unbroken and broken supersymmetry are investigated by the quantization condition of a method, called Asymptotic Iteration Method (AIM). Energy eigenvalues of the supersymmetric partner potentials are obtained. The present results are found to be in excellent agreement with numerical values. It is also noted that the AIM condition preserves the supersymmetric energy degeneracy, and generally fewer iteration steps are necessary to obtain the energy eigenvalues of the V+ (x) “fermionic” potential with the same accuracy. Key words: Asymptotic iteration method, supersymmetric quantum mechanics, partner potentials. PACS: 03.65.-w, 03.65.Ge, 11.30.Pb, 03.65.Db.
1. INTRODUCTION
Obtaining the exact solutions of Schr¨odinger equation is an interesting problem in fundamental quantum mechanics for lecturers, advanced undergraduate and graduate students in physics and applied mathematics. Unfortunately the range of exactly solvable potentials is limited. Consequently, there exist several methods to obtain approximate solutions of them, e.g. WKB approximation, time-independent perturbation theory [1], the numerical shooting method [2], the finite-element method [3,4], etc.. One of the methods mentioned above, the WKB approximation, is for solving the Schr¨odinger equation by expanding the wave function in the power series of ~. Although it is widely used in quantum mechanics and in many other branches of theoretical physics such as the theory of graded-index optical waveguides [5], the problem of exactness of the WKB approximation has arisen and various refinements have been developed to improve the accuracy of the method [6]. On the other hand, the application of supersymmetric quantum mechanics (SUSYQM) to bound-state problems has been the subject of extensive investigation in recent years [7–10]. In RJP 57(Nos. Rom. Journ. Phys., 3-4), Vol. 582–593 57, Nos. 3-4, (2012) P. 582–593, (c) 2012-2012 Bucharest, 2012
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SUSYQM a quantum mechanical Hamiltonian (H − ) is factorized by using differential operators and it is associated with a SUSY partner Hamiltonian (H + ). This partner Hamiltonian H + , can again be factorized, and a further new partner Hamiltonian is obtained. If a hierarchy of Hamiltonians can be constructed, one can also obtain a relation among the energy eigenvalues and the eigenfunctions of these Hamiltonians. If the ground-state energy of the initial potential is zero, then it can easily be written in factorisable form. Thus the ground-state energy of the partner Hamiltonian H + will be the energy of the first excited state of the initial one, H − . As a result, the differential operators in factorization act as raising and lowering operators for the eigenfunctions of these Hamiltonians. If the ground-state energy eigenvalue of the initial Hamiltonian is zero, then it has unbroken supersymmetry. Applying the differential operators to the excited state wavefunctions of the initial Hamiltonian one can obtain any wavefunction of the partner Hamiltonian. Finally, each of the new Hamiltonians will have one fewer bound-state, so that this process can be continued until the number of bound-states is exhausted. On the other hand, if the ground state energy of the initial potential is not zero, then it has broken supersymmetry and the operation of differential operators on the wavefunctions of Hamiltonians does not change the number of nodes of the partner wavefunctions (see Ref. [11] for details). Witten has introduced SUSYQM as a laboratory to test supersymmetry breaking [7], however, it is now widely studied in various other contexts. In addition to these applications, Comtet et al. formulated a supersymmetric version of the WKB method (SWKB) and demonstrated that SWKB can give exact energy eigenvalues for several solvable potentials with unbroken supersymmetry [12]. The method proposed by Comtet et al. was applied to some exactly solvable potentials [13] (for which the exact bound-state spectrum is reproduced) as well as to some non-exactly solvable models [14]. Later it was observed that the SWKB method yields the exact boundstate spectra [15] only for some shape-invariant potentials [16], and it was also found that the method fails to reproduce the energy levels for some non-shape-invariant potentials [17–19]. On the other hand, Dutt et al. proposed an alternative quantization condition known as broken supersymmetric WKB (BSWKB) for quantum mechanical bound-state problems with broken supersymmetry [20]. Unfortunately, it failed to give exact results in good agreement for broken supersymmetry. Recently a technique called the asymptotic iteration method (AIM) has been introduced [21] to obtain eigenvalues of second-order homogeneous differential equations. In the case of the Schr¨odinger equation the AIM was found to reproduce the energy spectrum exactly for most exactly solvable potentials [22–26], while for nonexactly solvable potentials it produces very good results [27–30]. In this study we apply AIM to some quantum mechanical problems in the framework of SUSYQM. First we briefly present an overview of AIM. Then we obtain the energy eigenvalues of some non shape-invariant potentials with unbroken and broken supersymmetry. RJP 57(Nos. 3-4), 582–593 (2012) (c) 2012-2012
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Finally we present our conclusions. 2. ASYMPTOTIC ITERATION METHOD
In this paper we apply AIM [21,22] to solve the second-order differential equations in the form f 00 (x) = λ0 (x)f 0 (x) + s0 (x)f (x) (1) 0 00 where f (x) is a function of x, f (x) and f (x) are the first and second derivatives with respect to x. λ0 (x) and s0 (x) are arbitrary functions in C∞ (a,b) and λ0 (x) 6= 0. To find a general solution, (1) can be iterated up to the (k + 1)th and (k + 2)th derivatives, where k = 1, 2, 3, ... is the iteration number. Then one obtains f (k+1) (x) = λk−1 (x)f 0 (x) + sk−1 (x)f (x) f (k+2) (x) = λk (x)f 0 (x) + sk (x)f (x),
(2)
where λk (x) = λ0k−1 (x) + sk−1 (x) + λ0 (x)λk−1 (x) sk (x) = s0k−1 (x) + s0 (x)λk−1 (x),
(3)
which are called the recurrence relation of (1). Taking the ratio of the (k + 2)th and (k + 1)th derivatives we get d f (k+2) (x) ln[f (k+1) (x)] = (k+1) dx f (x) λk (x) [f 0 (x) + (sk (x)/λk (x))f (x)] = . λk−1 (x) [f 0 (x) + (sk−1 (x)/λk−1 (x))f (x)]
(4)
Assuming that for sufficiently large k sk (x) sk−1 (x) = = α(x) λk (x) λk−1 (x)
(5)
holds, which is the “asymptotic aspect” of the method, (4) reduces to d λk (x) ln[f (k+1) (x)] = . dx λk−1 (x)
(6)
Substituting λk (x) from (3) and then using α(x) in the right handside of (6) we obtain Z λk (x) (k+1) f (x) = C1 exp dx λk−1 (x) Z x (7) [α(x1 ) + λ0 (x1 )]dx1 , = C1 λk−1 (x) exp RJP 57(Nos. 3-4), 582–593 (2012) (c) 2012-2012
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in which C1 is the integration constant. Inserting Eq.(7) into Eq.(2) and solving for f (x), we obtain the general solution of Eq.(1) as Z x f (x) = exp − α(x1 )dx1 × Z x1 Z x exp (λ0 (x2 ) + 2α(x2 ))dx2 dx1 . (8) × C2 + C1 The energy eigenvalues can be determined by the quantization condition given by the termination condition in Eq.(5). Thus, one can write the quantization condition combined with Equation (3) as δk (x) = λk (x)sk−1 (x) − λk−1 (x)sk (x) = 0,
k = 1, 2, 3, . . . .
(9)
Since fundamental quantum mechanics in general deals with the solution of the time-independent one-dimensional Schr¨odinger equation ~2 00 ψ (x) + [V (x) − E] ψ(x) = 0 , (10) 2m it is seen that the direct application of the method to Eq.(10) is not convenient. Therefore, one should rewrite wave function as −
ψ(x) = g(x) f (x)
(11)
(together with a possible change of coordinate) in order to transform Eq. (10) to the form (1). Then one can find analytically (or numerically) the energy spectrum and wave functions of the given potential. Although the general solution of Eq.(1) is given by Eq.(8), it is observed that the first part of Eq.(8) gives the polynomial solutions that are convergent and physical, whereas the second part gives non-physical solutions that are divergent. Therefore, C1 = 0 in Eq.(8) and the wave functions are determined by using the following wave function generator Z x sk (x0 ) 0 f (x) = C2 exp − dx . (12) λk (x0 ) On the other hand, Eq. (5) implies that the wave functions are truncated for sufficiently large values of k and the roots of the relation given in (9), which has been obtained from (3), belong indirectly to the spectrum of Eq.(10). However, for each iteration the expression (9) will depend on different variables, such as E, x and possible potential parameters. It is also noticed that the iterations should be terminated by imposing the quantization condition δk (x) = 0 as an approximation to (5) to obtain the eigenenergies. Therefore, the calculated eigenenergies En by means of this condition should be independent of the choice of the coordinate. The energy eigenvalues can easily be obtained from the roots of Eq. (9) if the problem is exactly solvable. If not, for a specific n quantum number, one has to choose a suitable x0 RJP 57(Nos. 3-4), 582–593 (2012) (c) 2012-2012
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point (which satisfies δk (x0 ) = 0), determined generally as the maximum value of the asymptotic wave function or the minimum value of the potential [21, 31, 32], and then the approximate energy eigenvalues are obtained from the roots of this equation for sufficiently large values of k with iteration. Thus the choice of x0 is observed to be critical only to the speed of the convergence of the eigenenergies, as well as for the stability of the process. It should be noted that exact solutions can be obtained whenever (1) can be transformed into the differential equation of some special function of mathematical physics. This is possible for specific choices of λ0 (x) and s0 (x), as discussed in Ref. [33]. The method was also combined with supersymmetric quantum mechanics to obtain the exact solution of some potentials [34]. 3. APPLICATIONS
Supersymmetric partner Hamiltonians [11] are defined in terms of the factorized form H − = A† A, H + = AA† , (13) † where A and A are each other’s Hermitian conjugates. This construction guarantees that the discrete spectrum of the SUSY partner potentials is non-negative. In the most frequently applied realization of SUSYQM A and A† are first-order differential operators ~ d ~ d A= √ + W (x), A† = − √ + W (x) , (14) 2m dx 2m dx where W (x) is called the superpotential. Then the two partner Hamiltonians correspond to one-dimensional Schr¨odinger operators ~2 d2 ~ + W 2 (x) ± √ W 0 (x) 2m dx2 2m ~2 d2 =− + V± (x) , 2m dx2
H± = −
(15)
where ~ W 0 (x) 2m are called the supersymmetric partner potentials. It is easily proven that V± = W 2 (x) ± √
H + (Aψ − ) = (AA† )Aψ − = E − (Aψ − ), H − (A† ψ + ) = (A† A)A† ψ + = E + (A† ψ + ),
(16)
(17)
i.e. the two Hamiltonians share the same spectrum, except possibly for the zeroenergy ground state, furthermore, the A and A† operators connect the iso-energetic levels of H − and H + . RJP 57(Nos. 3-4), 582–593 (2012) (c) 2012-2012
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The nature of the solutions ψ ± plays a special role in the interpretation of SUSYQM. Aψ − = 0 implies H − ψ − = 0, which means that ψ − is the ground-state energy of H − with E0 = 0 and ψ − = ψ0− is the ground-state wavefunction of H − and it is related to the superpotential by 0
W (x) = − √
~ (ψ0− (x)) . 2m ψ0− (x)
(18)
Since the ground-state wavefunction has to be nodeless inside its domain of definition, the partner potentials in Eq. (15) will not be singular there. In this case the first equation of (17) implies that the discrete spectrum of H + will not contain E = 0, so − En+1 = En+
(19)
will hold. This case corresponds to unbroken supersymmetry. It is also possible to consider solutions for which Aψ − 6= 0 holds. Obviously, this solution belongs to non-zero eigenenergy of H − and is generally non-normalizable. In this case the energy spectra of H − and H + is identical En− = En+
(20)
and this situation corresponds to “broken supersymmetry”. In what follows we consider certain potentials written in the supersymmetric form (16) and study their bound-state spectra in the case of both unbroken and broken supersymmetry by applying the quantization condition (9) and AIM. In what follows we shall use the units ~2 = 2m = 1, which does not affect the generality of our results. As a first application we analyze the one-dimensional anharmonic oscillator that is a non shape-invariant potential with unbroken supersymmetry [17, 35]: since the superpotential is defined as √ V0 x 3 W (x) = , (21) 3 a the partner potentials are found to be V0 x 6 1 p x 2 V± (x) = ± V0 . (22) 9 a a a As seen in 22, V− (x) is a symmetric double well potential, whereas the partner potential V+ (x) is a single well one. In order to apply the AIM, the Schr¨odinger equation for the partner potentials V± (x) V 0 x 6 1 p x 2 ± 00 −ψ (x) + ± V0 −E ψ(x) = 0, (23) 9 a a a has to be transformed into Eq. (1). Taking V− (x) in Eq.(23) with V0 = 1 and a = 1, RJP 57(Nos. 3-4), 582–593 (2012) (c) 2012-2012
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Table 1. − The fifth excited state (En=5 ) of the anharmonic potential V− (x) in Eq.(22) by means of the AIM and
nearly optimum values of β with V0 = 1 and a = 1. k 10 20 30 40 50 60 70 80 90 100
β=2 21.07006 15.52622 14.59486 14.58479 14.57981 14.58100 14.58088 14.58077 14.58076 14.58075
β=3 28.88411 17.78676 15.02977 14.60600 14.58137 14.58076 14.58075 14.58075 14.58075 14.58075
β=4 36.20133 20.90185 16.22441 14.83927 14.60119 14.58164 14.58078 14.58075 14.58075 14.58075
we consider a generic wave function g(β, x) = e−x
ψ(x) = g(β, x) f (x),
4 /12−βx2 /2
(24)
that satisfies the appropriate boundary condition: In the limit of large x the asymptotic solutions of Eq.(23) can be taken as any power of x times a decreasing Gaussian function, where β is the adjustment parameter. Substituting (24) in the eigenvalue equation (23) one obtains 2 2 2x2 00 0 2 f (x) = (x + 3β)x f (x) + β − E − β( + β)x f (x). (25) 3 3 Now comparing (25) and (1) one finds 2 λ0 (x) = (x2 + 3β)x, 3
s0 (x) = β − E − β(
2x2 + β)x2 . 3
(26)
Table 2. The first few energy eigenvalues of the anharmonic potential V± (x) [Eq.(22)] with V0 = 1 and a = 1, (−)
(+)
determined by the numerical method, the SWKB method and the AIM. The relation En+1 = En
is
also satisfied by AIM (where E0− = 0 for V− (x)). n− 1 2 3 4 5
Numerical [17] 1.1175 3.6364 6.7440 10.417 14.581
SWKB [17] 1.2566 3.6363 6.7453 10.417 14.581
AIM (V± ) 1.11745 3.63644 6.74401 10.4169 14.5808
n+ 0 1 2 3 4
It is now possible to calculate λk (x) and sk (x) applying Eq. (3). Finally, one finds the energy eigenvalues of the potential in (22) by using the quantization conRJP 57(Nos. 3-4), 582–593 (2012) (c) 2012-2012
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dition given in (9). In order to improve the energy eigenvalues, either the iteration number has to be increased or the most convenient adjustment parameter has to be determined phenomenologically. It turns out that the adjustment parameter β has a critical importance for the stability and the speed of convergence in this calculation. We simply try a set of values with β = 1, 2, ... for E5− and keep the ones that appeared to yield the best convergence rate (up to 7 exact digits) as seen in Table 1. We search for the best choice of the β value and the convergence seems to take place when k > 60 iterations, hence, in this work we choose that the minimum iteration number (k = 66) occurs when β = 3. For higher or lower values of β the iteration number increases dramatically for that energy eigenvalue. On the other hand, if one searches for the higher excited energy eigenvalues with fixed β = 3, then the iteration number k should be increased gradually, too. Therefore, we have set β = 3 and determined the first few eigenvalues (up to E5− ) in our calculations. Since the calculated eigenvalues En± should not depend on the x value applied in the condition (9), we observed that the optimal choice for x is at the maximum value of the asymptotic wave function, i.e. x = 0. Therefore, we also set x = 0 at the end of the iterations. We presented the energy eigenvalues determined by AIM in Table 2 together with the numerical and the SWKB values. We presented the AIM results up to 6 digits. Our results are in agreement with the numerical results. This is not true for SWKB approximation for the first few states. In addition to these calculations, we also obtain the energy eigenvalues for the partner potential V+ (x) via the AIM quantization condition by using the same generic wave function expression given in (24). We should note here that the eigenvalues of the partner potential V+ (x) are found after k = 52 iterations, − between two which is less than that for V− (x). The energy hierarchy, En+ = En+1 partner potentials it can be observed comparing the first and the fifth columns in Table 2, where we show the quantum states as n− and n+ for partner potentials V− (x) and V+ (x), respectively. It is seen that AIM can be an effective tool to determine the eigenvalue problem for such potentials if one properly determines the adjustment parameter, β. As a second application we consider the analytically unsolvable polynomial potential with unbroken supersymmetry [19]. The superpotential is now W (x) = 2bx + 4cx3 ,
(27)
V± (x) = (4b2 ± 12c)x2 + 16bcx4 + 16c2 x6 ± 2b
(28)
which results in
after substitution in Eq.(16). Before applying the quantization condition (9) to the potentials in Eq.(28), the eigenvalue equation has to be transformed again to Eq.(1). For this purpose we propose the wave function, satisfying the asymptotic behavior as x → ∓∞, in the form RJP 57(Nos. 3-4), 582–593 (2012) (c) 2012-2012
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Table 3. Energy eigenvalues of the potential V± (x) [Eq.(28)], with b=0.60, c=0.10 and β = 2.6, obtained by numerical method, the SWKB method and the AIM. n− 1 2 3 4 5 6 7 8
Numerical [19] 3.06619 7.19498 12.04113 17.48396 23.4480 29.8807 36.7429 44.0040
SWKB [19] 3.12274 7.24206 12.08354 17.52287 23.4841 29.9146 36.7750 44.0344
AIM 3.066187 7.194983 12.04113 17.48396 23.44796 29.88068 36.74292 44.00395
n+ 0 1 2 3 4 5 6 7
of 2
4
ψ(x) = e−βx −cx f (x), (29) where β is again adjustable parameter. Substituting (29) in the Schr¨odinger equation for the potentials in (28), it yields the following equation for f (x): f 00 (x) = 4 β + 2cx2 xf 0 (x) + 2β ± 2b − E + 4(b2 − β 2 + 6qc)x2 + 16c(b − β)x4 f (x). (30) Here q = 0 and q = 1 is used for V− (x) and V+ (x), respectively. Comparing Eq.(30) and Eq.(1), one obtains λ0 (x) = 4(β + 2cx2 )x, s0 (x) = 2β ± 2b − E + 4(b2 − β 2 + 6qc)x2 + 16c(b − β)x4
(31)
We can now calculate λk (x) and sk (x) by means of Eq. (3), and then we can find the energy eigenvalues of the potential, for example, V− (x) in (28) by using the quantization condition (9). The minimum of V− (x) depends on the parameters b and c such that for b2 > 3c the potential has one minimum [19]. Therefore we select b = 0.60 and c = 0.10. Furthermore, following the same procedure as in the previous example, the adjustment parameter observed to be optimal for β = 2.6. At the end of the iterations we again set x = 0, which is the maximum value of the asymptotic wave function. The minimum iteration number was found to be k = 74 for V− (x). The AIM results for V± (x), together with those from numerical and SWKB studies are shown in Table 3. The AIM results are displayed up to 7 digits. The number of iterations required to reach the same accuracy for the partner potential V+ (r) was found to be k = 68. The third example is a non-shape-invariant radial potential with broken supersymmetry. Dutt et al. [20] considered the superpotential W (r) = −2/r − r2 , RJP 57(Nos. 3-4), 582–593 (2012) (c) 2012-2012
(32)
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which leads to the SUSY partner potentials V− (r) =
2 6 + 6r + r4 and V+ (r) = 2 + 2r + r4 . r2 r
(33)
Considering the ansatz 1
ψ(r) = r2 e− 2 αr
2 − 1 βr 3 3
f (r).
(34)
for the wave functions of V− (r) (where α and β are adjustment parameters to be determined by AIM) one obtains f 00 (r) = 2βr2 + 2αr − 4/r f 0 (r) + 5α − E + 6(1 + β)r − α2 r2 − 2αβr3 + (1 − β 2 )r4 f (r). (35)
Table 4. Comparison of energy eigenvalues for the partner potentials derived from the spherically symmetric superpotential W (x) [Eq.(32)] by the numerical method, the BSWKB method and the AIM. Here En+ = En− holds. n± 0 1 2 3 4 5 6 7
Numerical [20] 13.34 23.39 34.58 46.74 59.73 73.45 87.84 102.83
BSWKB [20] 13.46 23.46 34.64 46.78 59.76 73.48 87.86 102.86
AIM (V± ) 13.3387 23.3854 34.5831 46.7396 59.7258 73.4483 87.8362 102.833
Comparing (35) and (1) one finds that λ0 (r) = 2βr2 + 2αr − 4/r, s0 (r) = 5α − E + 6(1 + β)r − α2 r2 − 2αβr3 + (1 − β 2 )r4 .
(36)
Again, using Eq.(3), we can compute λk (r) and sk (r) and determine the energy eigenvalues of V± (r) in (33) by using the quantization condition (9). The optimal choice for the adjustment parameters was now found to be α = 4 and β = 1/8, while for r0 we set r0 = 0.6695, where the maximum the asymptotic wave function is located. Table 4 contains the AIM results for both partner potentials, together with the numerical and WKB results. We displayed the AIM results up to 6 digits. It is also noted that our results which agree with numerical ones are obtained for k = 80 iterations for the potential V− (r), while for the same accuracy k = 78 was needed for the partner potential V+ (r). RJP 57(Nos. 3-4), 582–593 (2012) (c) 2012-2012
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4. CONCLUSION
In this study we applied the AIM to some analytical unsolvable supersymmetric partner potentials with unbroken or broken supersymmetry. We compared our results with the results of numerical and SWKB/BSWKB methods, and showed that the AIM can be directly applied to such problems if one introduces a wave function form that satisfies appropriate boundary conditions boundary conditions at zero and at infinity. We obtained accurate energy eigenvalues by using the quantization condition of the method. It was found that the AIM requires less iterations when applied to the partner potential V+ in this study. The largest difference occurs in the case of the first example, where the shape of V− (x) and V+ (x) is essentially different, while the iteration numbers are closest in the case of broken supersymmetry, where the potential shapes are similar and the number of bound states is the same. REFERENCES 1. L. I. Schiff, Quantum Mechanics, 3rd edn. (New York, McGraw-Hill, 1968). 2. N. J. Giordano, Computational Physics, (Englewood Cliffs, NJ Prentice-Hall, 1997). 3. L. R. Ram-Mohan, Finite Element and Boundary Element Applications in Quantum Mechanics (Oxford, Oxford University Press, 2002). 4. A. Ka-oey, Finite element treatment of anharmonic oscillator problem in quantum mechanics (BSc Dissertation, Naresuan University, 2004). 5. R. Srivastava, C. K. Kao, R.V. Ramaswamy, J. Lightwave Technol. 5, 1605 (1987). 6. F. Xiang, G. L. Yip, J. Lightwave Technol. 12, 443 (1994); S.V. Popov, B.M. Karnakov, V.D. Mur, Phys. Lett. A 210, 402 (1996); S. Zivanovic, V. Milanovic, V. Ikonic, Phys. Status Solidi B 204, 704 (1997). 7. E. Witten, Nucl. Phys. B 185, 513 (1981). 8. C. V. Sukumar, J. Phys. A 18, 2917 (1985); D. Baye, Phys. Rev. Lett. 58, 2738 (1987); D. Baye, J. Phys. A 20, 55529 (1987); F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251, 267 (1995); B. ¨ G¨on¨ul, O. Ozer, Y. Canc¸elik, M. Koc¸ak, Phys. Lett. A 275, 238 (2000). 9. G. L´evai, J. Phys. A 22, 689 (1989); G. L´evai, D. Baye, J-M. Sparenberg, J. Phys. A 30, 8257 (1997). 10. D. G´omez-Ullate, N. Kamran, R. Milson, J. Phys. A 37, 10065 (2004). 11. F. Cooper, A. Khare, U. Sukhatme, Supersymmetry in Quantum Mechanics (Singapore, World Scientific, 2001). 12. A. Comtet, A. Bandrauk, D. Cambell, Phys. Lett. B 150, 159 (1985). 13. A. Khare, Phys. Lett. B 161, 131 (1985). 14. R. Dutt, A. Khare, U. Sukhatme, Phys. Lett. B 181, 295 (1986). 15. R. Dutt R, A. Khare and U. Sukhatme, Am. J. Phys. 59, 723 (1991). 16. L.E. Gendenshtein, I.V. Krive, Sov. Phys. Usp. 28, 645 (1985). 17. R. Adhikari, R. Dutt, A. Khare, U. Sukhatme, Phys. Rev. A 38, 1679 (1988). 18. A. Khare, Y.P. Varshni, Phys. Lett. A 142, 1 (1989). 19. Y.P. Varshni, J. Phys. A 25, 5761 (1992).
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