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Brazilian Journal of Physics - The Hierarchy of Hamiltonians for a Restricted Class of Natanzon Potentials

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Brazilian Journal of Physics

Print version ISSN 0103-9733

Braz. J. Phys. vol.31 no.2 São Paulo June 2001

http://dx.doi.org/10.1590/S0103-97332001000200029 

The Hierarchy of Hamiltonians for a Restricted Class of Natanzon Potentials

 

Elso Drigo Filhoa*  and  Regina Maria Ricottab  
a
Instituto de Biociências, Letras e Ciências Exatas, IBILCE-UNESP
Rua Cristovão Colombo, 2265, 15054-000 São José do Rio Preto, SP, Brazil
b Faculdade de Tecnologia de São Paulo, FATEC/SP-CEETPS-UNESP
 
Praça Fernando Prestes, 30, 01124-060, São Paulo, SP, Brazil

 

Received on 10 November, 2000. Revised version received on 10 January, 2001

 

The restricted class of Natanzon potentials with two free parameters is studied within the context of Supersymmetric Quantum Mechanics. The hierarchy of Hamiltonians and a general form for the superpotential is presented. The first members of the superfamily are explicitly evaluated.

 

I  Introduction

The classes of Natanzon potentials, namely, the hypergeometric and the confluent, reffer to potentials whose Schrödinger equation is analytically and exactly solvable by means of hypergeometric functions. They have motivated several works concerning the mathematical and algebraic aspects of their structure and solutions and have numerous applications in several branches of physics, [1]-[7].

In particular, there have been studies within Supersymmetric Quantum Mechanics formalism. Cooper et al, [6], for instance, investigated the relationship between shape invariance and exactly analytical solvable potentials and showed that the Natanzon potential is not shape invariant although it has analytical solutions for the associated Schrödinger equation. Lévai et al, [7], have determined phase-equivalent potentials for a class of Natanzon potentials employing the formalism of supersymmetry.

However, the hierarchy of the Hamiltonians corresponding to Natanzon potentials has not been determined yet. In ref. [6], Cooper et al have sketched the first few potentials of the hierarchy from the knowledge of their asymptotic behaviour from the series approximation. In this paper we construct the hierarchy of Hamiltonians of the restricted class of Natanzon potentials, (Ginocchio class), with two free parameters. The first few members of the superfamily are explicitly evaluated and a general form for the superportential is proposed by induction.

 

II   Supersymmetric Quantum Mechanics Formalism

In the formalism of Supersymmetric Quantum Mechanics there are two operators Q and Q+, that satisfy the algebra 

where HSS is the supersymmetric Hamiltonian. The usual realisation of the operators Q and Q+ is 

where s± are written in terms of the Pauli matrices and A± are bosonic operators. With this realisation the supersymmetric Hamiltonian HSS is given by 

where H± are supersymmetric partner Hamiltonians and share the same spectra, apart from the nondegenerate ground state. Using the super-algebra a given Hamiltonian can be factorized in terms of the bosonic operators. In = c = 1 units, it is given by 

where E is the lowest eigenvalue. The bosonic operators are defined by

where the superpotential W1(r) satisfies the Riccati equation

The eigenfunction for the lowest state is related to the superpotential W as

or conversely

Now it is possible to construct the supersymmetric partner Hamiltonian,

If one factorizes H2 in terms of a new pair of bosonic operators, A one gets,

where E is the lowest eigenvalue of H2 and W2 satisfy the Riccati equation, 

Thus a whole hierarchy of Hamiltonians can be constructed , with simple relations connecting the eigenvalues and eigenfunctions of the n-members, [8]-[13]

 

III   Natanzon Potential and the Hierarchy of Hamiltonians

The restricted class of Natanzon potentials having two parameters and given in terms of the variable y(r) is,

where the variable function y(r) satisfies dy/dr = (1 - y2)[1 - (1 -l2)y2]. The dimensionless free parameters v and l measure the depth and the shape of the potential, respectively.

The Schrödinger equation for this potential, [2], [3], in dimensionless units, is given by 

where V(x) = v0 V(r), n = En/v0 and r = bx = (2mv0/2)1/2x .

The analytic solutions for the energy eigenfunctions are given by, 

where g(y) = 1 - (1 - l2)y2. The factor C(x) is a Gegenbauer polynomial when n is a non-negative integer, which is our case. The corresponding energy eigenvalues are given by n = - m l4, mn > 0, where 

Notice the relationship between the energy levels which will be extensively used in what follows,

In order to construct the superfamily we firstly factorize the Natanzon potential, calling V(r) = V1(r) = V-(r) + , [6]. The factorized Schrödinger equation is given by 

where = n. The superpotential W1(r) is evaluated from the knowledge of the ground state eigenfunction of V(r) by using (8) with Y = Yn , given by (18). It satisfies the Riccati equation and it is given by 

The superpartner Hamiltonian satisfies the equation 

which is written in terms of V2(r) as 

where V2(r), the potential for the second member of the hierarchy, is given by

To construct the next member of the superfamily, we factorize the Schrödinger equation for V2. It gives 

where W2(r) satisfies the associated Riccati equation, 

is the energy ground state of the potential V2(r) and it is such that = . The superpotential W2 can be computed from the ground state wave function Y . It is given by W2(r) = - log(Y), where Y = a1-y, i.e., 

where

and the coefficient a11 is given by 

The new superpartner of H2 is given by 

where V3(r), the potential for the third member of the hierarchy, is given by

and g(y) = 1 - (1- l2) y2. Thus, factorizing the Hamiltonian for this potential we have 

where W3(r) satisfies the Riccati equation, 

is the energy ground state of the potential V3(r), with = = . Again, the superpotential W3 can be computed from the ground state wave function Y, defined by W3(r) = -log(Y), with Y = a2-a1-y. It is given by 

where 

with coefficients are given by 

For the next member of the superfamily, we show the result of the evaluation of the superpotential, W4(r) = - log(Y) with Y = a3- a2- a1-y3(1). It is given by 

where f1 and f2 are evaluated in (29) and (36) and f3 is set to 

with the coefficients given by 

Casting all the results we have so far for the hierarchy, the following nth-term for the superpotential is induced 

where fn(y) is a 2n-order polynomial of the form

We stress that since Wn+1 is a superpotential it checks the Riccati equation, 

where Vn+1(r) is the superpartner potential of Vn which satisfies 

We have therefore a recursive relationship between Wn+1 and Wn given by 

where = - l4 and = - m l4. After the substitutions we end up with the condition 

where f¢ = df / dr and f¢¢ = d2f / dr2.

Therefore, fn+1 can be determined from the knowledge of fn . In this way, the particular cases of n=1, n=2 and n=3 can be checked by inspection and the resulting functions f1, f2 and f3 perfectly agree with equations (29) , (36) and (38) respectively. Notice that the particular case when l = 1 trivially reduces the n-th term superpotential, equation (39) to Wn+1 = y mn , with mn = v - n, since all the f's become 1 once all the ain's are checked to reduce to zero. The related potentials of the hierarchy are then given by

This is known as the shape invariant Pöschl-Teller (PT) potential.

 

IV   Conclusions

The hierarchy of Hamiltonians is studied for the restricted class of Natanzon potentials, (Ginocchio class), with two parameters and a general form for the superpotential is proposed. The superalgebra drives us to the conclusion that the whole superfamily is a collection of exactly solvable Hamiltonians. The case l = 1 served as a check of our formulae and was shown to reduce the original potential to the Pöschl-Teller (PT) potential, known to be shape invariant.

As a final remark, the shape invariance concept introduced by Gedenshtein, [12], has motivated several discussions about the exactly solvable potentials. In ref. [8] there is a discussion about this subject which has recently been extended in [14] concerning potentials depending on n parameters . The Natanzon potential is not shape invariant in the usual sense, as most of the exactly solvable potentials are. However, for the restricted class analised here, it was possible to obtain a general form for the superpotential, as shown in the previous section.

The Hulthén potential without the potential barrier term is another example of an exactly solvable potential which is not shape invariant, but for which it is possible to determine a general expression for the superpotential in the hierarchy, [13].

 

References 

[1] G. A. Natanzon, Theor. Mat. Fiz. 38, 146 (1979).         [ Links ]

[2] J. N. Ginocchio, Ann. Phys. 152, 203 (1984).         [ Links ]

[3] J. N. Ginocchio, Ann. Phys. 159, 467 (1985).         [ Links ]

[4] P. Cordero and S. Salamó, J. Phys A: Math. Gen. 24, 5299 (1991);         [ Links ] P. Cordero and S. Salamó, Jafarizadeh;         [ Links ] C. Grosche, J. Phys A: Math. Gen. 29, 365 (1996);         [ Links ] C. Grosche, J. Phys A: Math. Gen. 29, L183 (1996);         [ Links ] S. Codriansky, P. Cordero and Salamó , Nuovo Cimento 112B, 1299 (1997);         [ Links ] R. Milson, Int. J. Theor. Phys. 37, 1735 (1998).         [ Links ]

[5] M. A. Jafarizadeh, A. R. Estandyari and H. Panaki-Talemi, J. Math. Phys. 41, 675 (2000).         [ Links ]

[6] F. Cooper, J. N. Ginocchio and A. Khare, Phys. Rev. D36, 2458 (1987).         [ Links ]

[7] G. Levai, D. Baye and J.-M. Sparenberg, J. Phys A: Math. Gen. 30, 8257 (1997).         [ Links ]

[8] F. Cooper, A. Khare and U. P. Sukhatme, Phys. Rep. 251, 267 (1995).         [ Links ]

[9] C. V. Sukumar, J. Phys. A: Math. Gen. 18, L57 (1985).         [ Links ]

[10] C. V. Sukumar, J. Phys. A: Math. Gen. 18, 2917 (1985).         [ Links ]

[13] E. Drigo Filho and R. M. Ricotta, Mod. Phys. Lett. A14, 2283 (1989).         [ Links ]

[12] L. Gedenshtein, JETP Lett. 38, 356 (1983).         [ Links ]

[13] E. Drigo Filho and R. M. Ricotta, Mod. Phys. Lett. A10, 1613 (1995).         [ Links ]

[14] J. F. Cariñena and A. Ramos, J. Phys. A: Math. Gen. 33, 3467 (2000).        [ Links ]

 

 

*Work supported in part by CNPq