PHYSICAL REVIEW A VOLUME 50, NUMBER 5 NOVEMBER 1994 Expanding the class of conditionally exactly solvable potentials A. de Souxa Dutra Uniuersidade Estadual Paulista Ca-mpus de Guaratingueta-DFQ, Auenida Dr. Ariberto Pereira da Cunha, 333, Codigo de Enderegamento Postal 12500-000, Guaratingueta, Sio Paulo, Brazil Frank Uwe Girlich Fachbereich Physik, Universitat Leipzig, Augustusplatz 10, 04109Leipzig, Germany (Received 30 December 1993) Recently a class of quantum-mechanical potentials was presented that is characterized by the fact that they are exactly solvable only when some of their parameters are fixed to a convenient value, so they were christened as conditionally exactly solvable potentials. Here we intend to expand this class by in- troducing examples in two dimensions. As a byproduct of our search, we found also another exactly solvable potential. PACS number{s): 03.65.Ca, 03.65.Ge, 02.90.+p Recently the existence of a whole class of exactly solv- able potentials in quantum mechanics was discovered [1]. Furthermore it was observed that these potentials obey a supersymmetric algebra [2]. In fact the potentials dis- cussed in Ref. [1] were also discussed independently in a previous work of Stillinger [3]. In that article the author did not perceive that such potentials were only the first representatives of an entire new class of potentials. What distinguishes these potentials from the usual exactly solv- able potentials is the fact that at least one of their param- eters must be fixed to a specific value, because if you do not do that the potentials no longer have exact solutions. On the other hand, they cannot be classified among the so called quasi exactly solvable potentials [4—10], because these have only a finite number of exactly solvable energy levels, and this happens only when the potential parame- ters obey some constraint equations, so that only some of the parameters are free, and the others are fixed in terms of them. The continuous interest in obtaining and classi- fying quantum-mechanical potentials is because of their importance as a basis for expanding more realistic cases, and also for use of their solutions to test numerical calcu- lations. Another possibility is that of trying to use them as a kind of theoretical laboratory, searching for new in- teresting physical features. Some of the examples that we will study below have, in fact, the interesting characteris- tic of representing a kind of semi-infinite charged string, because, as can be seen in Fig. 1, the potential is infinitely attractive along the x semiaxis. Here in this work we in- tend to expand the class of conditionally exactly solvable (CES) potentials. This is going to be done through the presentation of two examples in two dimensions. As a byproduct of our search for CES potentials, we found another exactly solvable potential, whose solution will also be presented. The potentials that will be considered here are and Vn(x, y) = X A arctan— 2/3 8 [arctan(y/x) ] + +Cgo [arctan(y /x) ] (lb) 20 V p where the function arctan(y/x) is defined in the principal branch. It can be observed that the first case above has an exact solution for any value of Go if 8 =0. Otherwise the case 8%0 has an exact solution for Go= —3irt /32@, and only for a restricted region of the space (x and y both bigger or both less than zero). The second potential has exact solutions only when go= —5trt /32@. In this case the solution is valid along all the x-y plane. It is not difBcult to verify that these po- tentials are singular along the positive semiaxis x, as can be seen from Fig. 1. So they represent a kind of "charged string. " V, (x,y) = 1 x +y B arctan(y /x ) [arctan(y /x }]' 2 Go+ +C [arctan(y /x) ] +D(x +y ) FIG. 1. Plot of the potential CES II with unitary parameters. 1050-2947/94/50(5)/4369(4)/$06. 00 50 4369 1994 The American Physical Society 4370 BRIEF REPORTS d + V, (u) —6' '4(u)=0 t 2p du where (C+K —iri /8p, ) with K being the separation constant and p the mass of the particle. The subscript e in the potential stands for its exact solvability. In the variable v we have d & + VCEs(u) Eg(v—) =0, 2p dU with VCEs(u) having two possibilities: 8 Go VcEs(u) = + + U (4a) The solution for these potentials can be obtained through a suitable change of coordinates: x =u cos(u), y =u sin(u) . After this change of variables, and their separation, the Schrodinger equation can be split into two others. In the variable u we have that like multivaluedness of the wave function. However, one can see that this transformation is typical of a stereo- graphic mapping, in this case of R onto 5 . This kind of compactification can be done provided that one has suit- able asymptotic conditions [11].In fact, this implies that one will have a finite wave function throughout variation of the variables, and particularly that at infinity it should vanish. This imposes the condition that the parameter D be nonzero, because along the positive x semiaxis the wave function will be singular at infinity if D =0. Fur- thermore, there is no periodicity of the wave function in the variables x and y, as we will see later. So we see that the original potentials were, in some cases, reduced to one-dimensional CES potentials, whose solutions were obtained previously. Now we proceed to the solution of the above mentioned possibilities. Case I for 8 =0 and Gu arbitrary The fi. rst example is that of an exactly solvable potential. In this case the po- tential V, (x,y) is mapped into a harmonic oscillator with centrifugal barrier in the variable u, and a Coulomb plus centrifugal barrier in the variable U. So we only need to identify the corresponding parameters, substitute them in the equation for the "energy, " which in the transformed equation is the parameter K, and solve it for the original eAergp and VCEs(u) = Av + +II 2/3 (4b) 8 A D (, ,'„„(a=0)=4CD- A' [2m+ I+a, ]' (2n +1), 1/2 The wave function can be obtained by returning %(u, u) =u ' 4(u)y(u) to the original variables. At this point, we shall observe that the transformation used leads us to a mapping into an angle variable u =arctan(y /x ), which can, in principle, create problems I where a& —= (I+8pGO/fi )'~ . It can be seen that the above spectrum is a kind of mixing of the Coulomb and harmonic oscillator spectra. The normalized wave func- tions are qI, B =0( ) 32pD f2 1/8 21'(n +1) 1(a, +n+1) ' 1/2 ' 1/2 AD $2 (X +y2) 01 /2 arctan— X (2~) + 1)/41/2 —SpK $2 I (m +1) I (a&+m+1) arctan-y X j . 1/2 arctan —expy pD 2%2 1/2—8@K (X2+y )— ( 1/2 AD ( z+ z) J 1/2—2pE arctan— $2 X where p 2A D 1/2 —(2n +1), (2n +1) —C, 2pA (2m +a, + I )iri 1/2 2 1 ( fi 2Dx=- (7b) 4C "' p p and L„'(x,y) is the generalized Laguerre polynomial. It is remarkable that, as can be seen from Eq. (5), the domain of validity of the parameter C is defined through t equation in order to keep the energies real. So, as the left hand side of the above equation has its minimum value when m =0 (n arbitrary), we must have C ~ 2@A /(a & +1)A' . Apparently if C is less than this we should impose the condition that where Int( ) stands for taking the first integer greater than or equal to the argument. This is a strange feature because, in this case, the ground state would have nodes looking like an excited state. This characteristic should 50 BRIEF REPORTS 4371 with ' 1/2 2A DK 1 (2n+1) —C . (11) p As we are looking for 8'„,we see that it is necessary to solve a third order polynomial equation for E, and then use this solution in the expression for the energy ' 1/2 2A D=+2&D (K +C)' + (2n +1); be verified through numerical calculation. Perhaps we would need some kind of analytical continuation for the energy in such cases. Now, the next case. Case Ifor 8%0 and Go= —3A' /32p. In this case we have our first example of a two-dimensional CES poten- tial. After repeating the previous procedure we get the following equation for the energy eigenstates: pB4i' (m+ —,')K +2pA K +pAB K + =0, (10) K =(R +D)' +(R D—)' a1 m 3 (13a) with D—:+Q +R, Q:—(3az —a & )/9, R:—(9a&a2—27a & —2a, ) /54, and 2@A pAB vari (m +—') fi (m +—') B4 8' (m+ —,') (13b) once more we have a limitation on the parameter C. The solution of the equation for the parameter E is obtained following the procedure appearing in Ref. [1]. This gives us (12) The wave function is given by ' 1/2('+') $2 ' 1/2' 1/8 tg (x,y) = 32p,D En+1) 2 + m!I (a+n+1) 1/2 AD B f2 (x +y ) H P &arctan(y/x)— 2EXL: B Xexp — z (x +y ) — &arctan(y/x)— 2E 2 a/2 ' 1/4 arctan- x (14) where we defined ' 1/2 —(2n +1), 2A2D and 2R D p 1/2 (2n + 1) —C, (15a) ' 1/2 1/2 +4v'A (m +1/2)iri +— 3 2 4p 8pK $2 ' 1/4 with the solution ' 1/2 1/2 29iri A + 98 n, m 2p 4 and H (x,y) is the Hermite polynomial. It is easy to verify that the wave function, being pro- portional to arctan(y/x), vanishes along the positive x semiaxis. This should be expected due to the singularity of the potential in this region. Moreover, it is easy to see from the wave function (14) that, as observed previously when we presented the potential, one must restrict the re- gion of space such that x and y are both positive or both negative. Furthermore, the inverse tangent function is defined in the principal branch, as said previously. Final- ly we treat the second case. Case II (go= —5R /72p). Now we present the last ex- ample of a two-dimensional CES potential that will be discussed in this work. If we repeat once more the same procedure we get the following equation for the energy in this last case: 1/2 1/2 +4CD + p (2n + 1) . (15b) B~— (16) in order to include m =0 in the solution. Besides, the pa- rameter C is also restricted by C) 4&A ( +, ) 9A' A 3 2p 1/2 1/2 (17) This time we need to do a careful analysis of the range of validity for the potential parameters. In the case of the parameter B, we have the restriction 1/2 BRIEF REPORTS So we see that the negative solution (positive in the above equation) introduces a limit on the number of bound states. The positive case (negative above) is such that we only need impose that C is bigger than the left hand side of Eq. (17) with m =0. After this brief analysis of the region of validity of the potential parameters, we present the corresponding wave functions: ] /8 (x,y) = 32 D tl, m 3r( +1) 2 'm!I (a +n +1) 2pa $2 (x +y ) ~ /2 2/3 ] 1/4 z~ arctan — P XL„' 1/2 (x'+y ) 8 ~P arctan— 2/3 pDX exp 2A ][ /2 (x+y ) —— 2 arctan—3' X E ' 2A where p 2A D —(2n +1) and ' ]/4 9@A 2A (19) This finishes the presentation of examples of CES poten- tials, now in two-dimensional space. The principal feature presented by these potentials, besides belonging to the CES class of potentials, is that a stringlike singularity of the potential appears. This is corroborated by the fact that the probability of finding the particle over this singu- larity is zero. This accounts for the interest in doing fur- ther investigations of the characteristics of such poten- I tials. It would be interesting to study the behavior of their corresponding coherent states, and also see what happens with wave packets around the linear singularity of the potential. Furthermore, in analogy with what has been done with the exact potentials [12,13], and also with the quasiexact ones [8,9], it is our intention to look for the dynamical algebra behind the solvability of the CES potentials. This study is presently under development and we hope to re- port on it in the near future. One of the authors (A.S.D.) thanks CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecnologico) of Brazil for partial financial support and FAP ESP (Fundal'Ko de Amparo a Pesquisa no Estado de Sao Pau- lo) for providing computer facilities. [1]A. de Souza Dutra, Phys. Rev. A 47, R2435 (1993). [2] E. Papp, Phys. Lett. A 17$, 231 (1993). [3] F. H. Stillinger, J. Math. Phys. 20, 1891 (1979). [4] G. P. Flessas, Phys. Lett. 72A, 289 (1979); 78A, 19 (1980); 81A, 17 (1981);J. Phys. 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