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334 Brazilian Journal of Physics, vol. 31, no. 2, June, 2001 The Hierarchy of Hamiltonians for a Restricted Class of Natanzon Potentials Elso Drigo Filhoa and Regina Maria Ricotta b a Instituto de Bioci^ encias, Letras e Ci^encias Exatas, IBILCE-UNESP Rua Cristov~ao Colombo, 2265, 15054-000 S~ao Jose do Rio Preto, SP, Brazil b Faculdade de Tecnologia de S~ ao Paulo, FATEC/SP-CEETPS-UNESP Praca Fernando Prestes, 30, 01124-060, S~ao 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 rst members of the superfamily are explicitly evaluated. I Introduction The classes of Natanzon potentials, namely, the hypergeometric and the conuent, reer to potentials whose Schrodinger 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 Schrodinger equation. Levai 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 rst 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 rst few members of the superfamily are explicitly evaluated and a general form for the superportential is Work supported in part by CNPq 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 + fQ; Qg = fQ + + ;Q g = 0; fQ; Q g = HSS + (1) where HSS is the supersymmetric Hamiltonian. The usual realisation of the operators Q and Q is 0 0 Q=a = A 0 + 1 = 00 A0 (2) where are written in terms of the Pauli matrices and A are bosonic operators. With this realisation the supersymmetric Hamiltonian HSS is given by A A 0 H 0 HSS = 0 A A = 0 H : (3) where H are supersymmetric partner Hamiltonians and share the same spectra, apart from the nondegenerate ground state. Using the super-algebra a given + Q =a + + 1 + + + + 335 Elso Drigo Filho e Regina Maria Ricotta Hamiltonian can be factorized in terms of the bosonic operators. In ~ = c = 1 units, it is given by H1 d2 = dr2 + V (r) = A +E + 1 A1 1 The eigenfunction for the lowest state is related to the superpotential W as (4) (1) 0 (r) = N exp( (1) 0 where E is the lowest eigenvalue. The bosonic operators are dened by (1) 0 = drd + W (r) A1 ( )= W1 r where the superpotential W (r) satises the Riccati equation 0 W W = V (r) E : (6) 1 2 1 (7) d dr ( ): (8) (1) 0 ln Now it is possible to construct the supersymmetric partner Hamiltonian, (1) 0 1 1 () ) W1 r dr 0 or conversely (5) 1 Z r c H2 =A 1 + A1 d2 +E = (1) 0 dr2 + (W + W 0 ) + E 2 1 1 (9) (1) 0 : If one factorizes H in terms of a new pair of bosonic operators, A one gets, 2 2 H2 =A + 2 A2 d2 +E = (2) 0 dr2 + (W 2 2 0 ) + E (2) W2 (10) 0 d where E is the lowest eigenvalue of H and W satisfy the Riccati equation, 0 W W = V (r) E : (11) Thus a whole hierarchy of Hamiltonians can be constructed , with simple relations connecting the eigenvalues and eigenfunctions of the n-members, [8]-[13] n Hn = An An + E (12) (2) 0 2 2 2 ( ) 0 + = An drd + Wn (r) + + 1 A2 ::: (1) (n+1) 0 (r) = N exp( (1) 0 (2) 0 2 2 n = A 2 (1) En Z r 0 =E n (14) () ) (15) ( +1) 0 W1 r dr : 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, (13) c ( ) = f v(v + 1) + 1=4(1 )[5(1 )y (7 )y + 2]g(1 y ) ; (16) where the variable function y(r) satises dy=dr = (1 y )[1 (1 )y ]. The dimensionless free parameters v and measure the depth and the shape of the potential, respectively. The Schrodinger equation for this potential, [2], [3], in dimensionless units, is given by V r 2 2 2 4 2 [ 2 d =dr 2 2 2 2 2 2 + V (r)]n (r) = n n (r) (17) where V (x) = v V (r), n = En =v and r = bx = (2mv =~ ) = x. The analytic solutions for the energy eigenfunctions are given by, 0 0 n = (1 0 2 )n = [g(y)] 2 2 1 2 (2n +1)=4 Cn n +1=2 (y=[g(y)] = ) 1 2 (18) 336 Brazilian Journal of Physics, vol. 31, no. 2, June, 2001 where g(y) = 1 (1 )y . The factor Cna (x) is a Gegenbauer polynomial when n is a non-negative integer, which is our case. The corresponding energy eigenvalues are given by n = n ; n > 0, where = n = [ (v + 1=2) + (1 )(n + 1=2) ] (n + 1=2): (19) Notice the relationship between the energy levels which will be extensively used in what follows, (n n ) = ( 2n (2n + 1)n + (2n 1)n ): (20) 2 ( ) 2 2 2 2 2 2 2 2 4 2 1 2 2 1 1 d In order to construct the superfamily we rstly factorize the Natanzon potential, calling V (r) = V (r) = V (r) + , [6]. The factorized Schr odinger equation is given by H = a a ; a = d=dr + W (r) (21) where n = n . The superpotential W (r) is evaluated from the knowledge of the ground state eigenfunction of V (r) by using (8) with n = n , given by (18). It satises the Riccati equation and it is given by W (r) = (1 )y (y 1)=2 + y : (22) (1) 0 (1) 0 1 The superpartner Hamiltonian satises the equation 1 + 1 1 1 1 (1) H2 1 2 0 1 (23) + a1 2 2 W1 (1) 2 =a which is written in terms of V (r) as 1 2 (1) 0 + W 0 = V (r) (24) (1) 0 2 1 where V (r), the potential for the second member of the hierarchy, is given by 2 c ( )=f 2 4 V2 r 0 + + 1=4(1 2 0 2 )[ 7(1 2 )y + (9 3 4 8 )y 2 0 2 2]g(1 2 y 2 ): (25) d To construct the next member of the superfamily, we factorize the Schrodinger equation for V . It gives H = a a ; a = d=dr + W (r) (26) where W (r) satises the associated Riccati equation, 0 W W = V (r) : (27) is the energy ground state of the potential V (r) and it is such that = . The superpotential W can be computed from the ground state wave function . It is given by W (r) = drd log( ), where =a , i.e., W (r) = 3=2(1 )y (y 1)+ y drd log(f ) (28) where ( ) =1+a 2 (2) 0 2 + 2 2 2 2 2 2 2 (2) 0 (2) 0 (2) 0 (2) 0 2 (1) 1 2 1 (1) 1 2 2 2 1 2 1 2 (29) ; 11 a11 = ( ) 1: 1 2 0 (30) The new superpartner of H is given by 2 (2) 0 2 11 y and the coeÆcient a is given by (2) 0 2 2 f1 y W22 + W 0 = V (r ) 3 2 (31) (2) 0 where V (r), the potential for the third member of the hierarchy, is given by 3 c ( ) = f + + 1=4(1 ) 27(1 )y + (33 15 )y 6 + + 2af (gy()y) f1 + ( 9 + 6 2 )y + 8(1 )y g + 8a y (1 y )( fg((yy)) ) g(1 2 1 V3 r 11 1 4 1 2 2 2 2 1 2 2 4 2 2 4 2 2 11 2 2 2 1 (32) y 2 ): 337 Elso Drigo Filho e Regina Maria Ricotta and g(y) = 1 (1 ) . Thus, factorizing the Hamiltonian for this potential we have H = a a ; a = d=dr + W (r) (33) where W (r) satises the Riccati equation, W W 0 = V (r ) : (34) is the energy ground state of the potential V (r), with = = . Again, the superpotential W can be computed from the ground state wave function , dened by W (r) = drd log( ), with = a a . It is given by W = 5=2(1 )y (y 1) + y + drd log(f ) drd log(f ) (35) where f (y ) = 1 + a y + a y (36) with coeÆcients are given by a = 2( ) 2 2 y 2 (3) 0 3 + 3 3 3 3 3 2 3 (3) 0 3 2 3 (3) 0 3 3 (3) 0 (3) 0 (2) 1 (1) 2 (3) 0 3 2 2 2 2 2 21 21 1 1 3 (3) 0 2 2 (4) 0 0 1 (4) 0 3 W4 = 7=2(1 2 4 2 1 4 22 2 2 0 1 2 0 2 0 2 0 2 )y(y 1) + y + drd log(f ) + drd log(f ) 2 2 3 1 d 2 dr where f and f are evaluated in (29) and (36) and f is set to f (y ) = 1 + a y + a y + a y ; with the coeÆcients given by a = 3( ) + 2( ) 5 1 1 1 2 (1) 3 2 (1) 2 2 = 1 + (2 3 + 6 )=3 + ( 4 + 6 2 5 + 3 + 3 )=3: For the next member of the superfamily, we show the result of the evaluation of the superpotential, d . It is given by dr log ( ) with = a a a a22 1 ( ) ()= W4 r (37) log f3 3 3 31 = 10 + + a32 2 31 2 2 3 4 32 0 (38) 6 33 2 1 (6 25 + 15 6 + 22 )=3 + (6 + 2 18 6 + 18 + 13 3 6 11 3 + 8 )=3 2 0 4 0 1 1 2 0 2 2 1 3 0 3 0 1 3 1 2 3 0 2 2 3 148 484 206 84 + ( 6 + 13 3 12 4 ) + + 15 45 45 5 42 1 394 497 59 16 + 45 + 2 + 3 45 + 45 5 3 + 116 24 48 + 12 + 8 + 82 + 83 + 328 45 45 5 5 5 3 9 14 6 + 2 8 58 + 182 2 9 15 45 45 5 26 + 8 + 44 4 5 9 9 Casting all the results we have so far for the hierarchy, the following nth-term for the superpotential is induced a33 = 10 + 2 0 0 6 Wn+1 1 2 2 0 1 2 0 2 1 3 = (n + 1=2)(1 2 0 2 1 3 )( 2 y y 2 1 2 0 3 0 0 3 2 2 2 0 ( )= n X i=0 ani y 2i ; an0 1 2 0 3 Qn where fn (y) is a 2n-order polynomial of the form fn y 1 3 1 3 2 3 2 0 2 1 3 3 1) + yn + drd log( 2 3 1 0 0 2 0 0 3 3 0 2 4 3 1 2 0 0 2 0 1 = 1: i=0 fi fn 1 ) ; f0 =f =1 1 (39) (40) 338 Brazilian Journal of Physics, vol. 31, no. 2, June, 2001 We stress that since Wn is a superpotential it checks the Riccati equation, +1 0 = Vn (r) n where Vn (r) is the superpartner potential of Vn which satises Wn + Wn0 = Vn (r) n : We have therefore a recursive relationship between Wn and Wn given by 2 Wn+1 Wn+1 (41) ( +1) 0 +1 +1 2 +1 (42) 0 +1 Wn0 +1 Wn2+1 where n = 0 = Wn + Wn0 + n 2 (43) (n+1) 0 0 and n = n . After the substitutions we end up with the condition 2n(1 ) y (y 1) + (y 1) ((2n 1)n (2n + 1)n )(1 (1 )y ) 2n + 2n 1 ( +1) 0 4 2 2 2 + nX1 fi0 f i=0 i +fn0 n nX1 2 2 fn0 1 n 1 4f 1 4 2 1 2 2 fn0 2 f + 2(1 n 2 1 2 )y(y 2 1) + 2y (n 2 2 fi00 f i=0 i 2 1 1 1 +2 2 2 2 nX1 i=0 2 n 4n(1 )y(y 1) 2fn0 =fn + 2y (n + n ) (1 )y(y 1)(2n + 1) + 2y n + fn00 =fn 1 =fn f 0 =fn 2 1 ) + (44) 1 2 0 ( ffi ) + 2n(1 i 1 2 1 2 )(1 3y ) 2 2 (n + n ) 1 dy=dr = 0: where f 0 = df =dr and f 00 = d f =dr . 2 2 d Therefore, fn 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 f , f and f perfectly agree with equations (29) , (36) and (38) respectively. Notice that the particular case when = 1 trivially reduces the nth term superpotential, equation (39) to Wn = yn , with n = 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 Vn (r) = (v n)(v n + 1)=cosh r n = 0; 1; ::: (45) This is known as the shape invariant Poschl-Teller (PT) potential. 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 Hulthen 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]. IV Conclusions References +1 1 2 3 +1 +1 2 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 = 1 served as a check of our formulae and was shown to reduce the original potential to the Poschl-Teller (PT) potential, known to be shape invariant. As a nal remark, the shape invariance concept introduced by Gedenshtein, [12], has motivated several [1] G. A. Natanzon, Theor. Mat. Fiz. 38, 146 (1979). [2] J. N. Ginocchio, Ann. Phys. 152, 203 (1984). [3] J. N. Ginocchio, Ann. Phys. 159, 467 (1985). [4] P. Cordero and S. Salamo, J. Phys A: Math. Gen. 24, 5299 (1991); P. Cordero and S. Salamo, Jafarizadeh; C. Grosche, J. Phys A: Math. Gen. 29, 365 (1996); C. Grosche, J. Phys A: Math. Gen. 29, L183 (1996); S. Codriansky, P. Cordero and Salamo , Nuovo Cimento 112B, 1299 (1997); R. Milson, Int. J. Theor. Phys. 37, 1735 (1998). Elso Drigo Filho e Regina Maria Ricotta [5] M. A. Jafarizadeh, A. R. Estandyari and H. PanakiTalemi, J. Math. Phys. 41, 675 (2000). [6] F. Cooper, J. N. Ginocchio and A. Khare, Phys. Rev. D36, 2458 (1987). [7] G. Levai, D. Baye and J.-M. Sparenberg, J. Phys A: Math. Gen. 30, 8257 (1997). [8] F. Cooper, A. Khare and U. P. Sukhatme, Phys. Rep. 251, 267 (1995). [9] C. V. Sukumar, J. Phys. A: Math. Gen. 18, L57 (1985). 339 [10] C. V. Sukumar, J. Phys. A: Math. Gen. 18, 2917 (1985). [11] E. Drigo Filho and R. M. Ricotta, Mod. Phys. Lett. A14, 2283 (1989). [12] L. Gedenshtein, JETP Lett. 38, 356 (1983). [13] E. Drigo Filho and R. M. Ricotta, Mod. Phys. Lett. A10, 1613 (1995). [14] J. F. Cari~nena and A. Ramos, J. Phys. A: Math. Gen. 33, 3467 (2000).