The sextic oscillator as a γ-independent potential

The sextic oscillator as a ␥ -independent potential G. Lévai* Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI), P.O. Box 51, H-4001 Debrecen, Hungary J. M. Arias† Departamento de Física Atómica, Molecular y Nuclear Facultad de Física, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain (Received 7 October 2003; published 20 January 2004) The sextic oscillator is proposed as a two-parameter solvable ␥ -independent potential in the Bohr Hamiltonian. It is shown that closed analytical expressions can be derived for the energies and wave functions of the first few levels and for the strength of electric quadrupole transitions between them. Depending on the parameters this potential has a minimum at ␤ =0 or at ␤ ⬎0, and might also have a local maximum before reaching its minimum. A comparison with the spectral properties of the infinite square well and the ␤ 4potential is presented, together with a brief analysis of the experimental spectrum and E2 transitions of the 134Ba nucleus. DOI: 10.1103/PhysRevC.69.014304 PACS number(s): 21.10.Re, 03.65.Ge I. INTRODUCTION In the last few years there has been considerable interest in looking for analytic solutions of the Bohr Hamiltonian which describes the collective motion in nuclei in terms of shape variables ( ␤ , ␥ )[1]. This was initiated with the introduction of the interacting boson model (IBM)[2]. This model presents four different dynamical symmetries, each one associated with a well-defined nuclear shape. The model Hamiltonian provides a natural way of going from a phase to another one by changing systematically few parameters and, consequently, allows one to study shape phase transitions. Thus, the IBM phase diagram has been analyzed from different points of view [3–7]. Specially important are the critical points since in these situations structural changes occur rapidly and it is difficult to design appropriate theoretical models. Recently, Iachello has proposed a new dynamical symmetry for describing the critical point at the transition from spherical to deformed ␥ -unstable shapes [8]. This symmetry has been called E共5兲and is expected to occur in nuclei in which the V共 ␤ , ␥ 兲potential depends only on the ␤ variable, and it has a relatively flat shape in the ␤ variable. Around this critical point of the phase transition the potential appearing in the Bohr Hamiltonian can be approximated by an infinite square well in the ␤ variable. In this case the Bohr Hamiltonian can be solved exactly in terms of Bessel functions, and various quantitative predictions can be obtained for various spectroscopic properties [ratios of the excitation energies and B共E2兲transitions], on the basis of which one can search for candidates for the E共5兲symmetry among nuclei. Lately, two other dynamical symmetries X共5兲[9]and Y共5兲[10]to describe the critical point at the transition from spherical to axially deformed shapes and from axially deformed to triaxial shapes, respectively, have been proposed. The introduction of the E共5兲symmetry renewed the interest in studying exactly solvable V共 ␤ , ␥ 兲potentials. Most efforts have concentrated on ␥ -independent potentials, for which the most well-known example is the harmonic oscillator in the five-dimensional space [11]and its extension which contains a term proportional to ␤ −2 [12]. More recently two more exactly solvable ␥ -unstable potentials have been discussed: the Coulomb and the Kratzer potentials [13]. Here again the latter potential is an extension of the former in the sense that it contains a term proportional to ␤ −2, which formally changes the ␶ variable, the analog of the langular momentum of radial potentials in three spatial dimensions. This formal changing of ␶ introduces a minimum of the potentials at ␤ ⬎0 in both cases [12,13]. The bound solutions of the harmonic oscillator and the Coulomb potentials (and their extensions)are given in terms of generalized Laguerre polynomials [14]. Similar to the three-dimensional case, these examples, together with the infinite square well, practically exhaust those exactly solvable potentials that contain a ␤ −2 term. Here we propose another potential which has this property, and although it is not exactly solvable in the classical sense, it has a number of features that make it an ideal potential to be used in the Bohr Hamiltonian. This is the sextic oscillator, which belongs to the class of quasi-exactly solvable potentials [15]. These potentials have the property that their solutions can be obtained in closed form for a number of energy eigenvalues, i.e., for the lowest few values of the nprincipal quantum number, which are also the lowest in energy. This is clearly sufficient for potentials appearing in the Bohr Hamiltonian. It is rarely necessary to consider more than a few levels with the same angular momentum, and these can be obtained from the lowest few solutions of the sextic oscillator. Furthermore, the sextic oscillator has a more flexible shape than other solvable potentials, as depending on its parameters, it can have a minimum at ␤ =0 or at ␤ = ␤ min⬎0, and in addition, it can also have a local maximum at ␤ max⬍ ␤ min. The paper is structured as follows. In Sec. II the Bohr Hamiltonian is revised together with its solutions for ␥ -independent potentials. In Sec. III the lowest energy solu- *Electronic address: [email protected] †Electronic address: [email protected] PHYSICAL REVIEW C 69, 014304 (2004) 0556-2813/2004/69(1)/014304(6)/$22.50 ©2004 The American Physical Society69 014304-1 tions of the Bohr Hamiltonian for the sextic oscillator potential are worked out. Section IV is devoted to show the simple use of the sextic oscillator as a flexible ␥ -independent potential. Finally, in Sec. V we present preliminary applications and discuss future extensions. II. THE BOHR HAMILTONIAN FOR ␥ -INDEPENDENT POTENTIALS Let us first consider that the Bohr Hamiltonian describing the collective motion of a deformed nucleus in the fivedimensional space determined by the ␪ iEuler angles 共i =1,2,3兲and the intrinsic ␤ and ␥ variables is [1] H=− ប2 2B 冢 1 ␤ 4 ⳵ ⳵ ␤␤ 4 ⳵ ⳵ ␤ +1 ␤ 2sin 3 ␥ ⳵ ⳵ ␥ sin 3 ␥ ⳵ ⳵ ␥ −1 4 ␤ 2兺 k Qk 2 sin2 冉 ␥ −2 3 ␲ k 冊 冣 +V共 ␤ , ␥ 兲.共1兲 In what follows we assume that the potential in Eq. 共1兲de- pends only on ␤ , i.e., V共 ␤ , ␥ 兲=U共 ␤ 兲. For these ␥ -independent potentials the wave functions can be separated into two parts, ⌿共 ␤ , ␥ , ␪ i兲=f共 ␤ 兲⌽共 ␥ , ␪ i兲,共2兲 which satisfy the following differential equations: 冢 −1 sin 3 ␥ ⳵ ⳵ ␥ sin 3 ␥ ⳵ ⳵ ␥ +1 4兺 k Qk 2 sin2 冉 ␥ −2 3 ␲ k 冊 冣 ⌽共 ␥ , ␪ i兲 =⌳⌽共 ␥ , ␪ i兲, ⌳= ␶ 共 ␶ +3兲, ␶ =0,1,2, ..., 共3兲 冉 −1 ␤ 4 ⳵ ⳵ ␤␤ 4 ⳵ ⳵ ␤ +⌳ ␤ 2+u共 ␤ 兲 冊 f共 ␤ 兲= ⑀ f共 ␤ 兲.共4兲 Here we have introduced ⑀ =共2B/ប2兲Eand u共 ␤ 兲 =共2B/ប2兲U共 ␤ 兲. Note that the ␶ values determine the allowed angular momenta J, too 关16兴. By setting ␾ 共 ␤ 兲= ␤ 2f共 ␤ 兲we obtain an equation which has the form of a radial Schrödinger equation −d2 ␾ d ␤ 2+ 冉 共 ␶ +1兲共 ␶ +2兲 ␤ 2+u共 ␤ 兲 冊 ␾ = ⑀␾ .共5兲 Note that this is different from Eq. 共6兲in Ref. 关8兴, in that it contains no linear derivative term due to the different definition of ␾ 共 ␤ 兲. This choice also implies that the factor corresponding to the ␤ volume element in the integration of functions of the type f共 ␤ 兲in Eq. 共4兲has been transferred to the solutions of the type ␾ 共 ␤ 兲in Eq. 共5兲. Thus, in the integration of these no factor arising from the volume element has to be included. The complete solution of the problem implies the solution of Eq. 共3兲, too; this was solved in Ref. 关16兴. III. THE SEXTIC OSCILLATOR The sextic oscillator with a centrifugal barrier is defined [15]as H=− d2 dx2+共2s− 1/2兲共2s− 3/2兲 x2+ 冋 b2−4a 冉 s+1 2+M 冊 册 x2 +2abx4+a2x6,共6兲 where x苸关0, ⬁兲and Mis a non-negative integer. This potential is quasi-exactly solvable, which means that for any non-negative integer value of M,M+1 of its solutions can be obtained in an algebraic way. The 共unnormalized兲solutions are written in the form ␾ n共x兲=Pn共x2兲共x2兲s−1/4exp 冉 −a 4x4−b 2x2 冊 ,n=0,1,2, ..., 共7兲 where Pnis a polynomial of order n. Obviously, normalizability requires a艌0, while a=0 reduces the problem to the exactly solvable harmonic oscillator. The simplest solutions are obtained for M=0 and M=1 [15]. For M=0 only one nodeless (i.e., ground state)solution appears at E0 共M=0兲=4bs, with the corresponding wave function being ␾ 0 共M=0兲共x兲⬃共x2兲s−1/4exp 冉 −a 4x4−b 2x2 冊 .共8兲 For M=1 two solutions appear, one nodeless and another with one node for x⬎0. These correspond to the ground state and the first excited state, respectively, at energies E0 共M=1兲 =4bs+␭−共s兲and E1 共M=1兲=4bs+␭+共s兲, where ␭±共s兲=2b±2共b2+8as兲1/2 共9兲 are the roots of the equation ␭2−4b␭−32as=0. The corresponding wave functions are ␾ n 共M=1兲共x兲⬃ 冉 1− ␭ 8sx2 冊 共x2兲s−1/4exp 冉 −a 4x4−b 2x2 冊 , 共10兲 and the ␭=␭−共s兲and ␭=␭+共s兲choice has to be made for n =0 and n=1, respectively 关15兴.关Note that a艌0 and s艌0 imply ␭−共s兲艋0, so the polynomial part of Eq. 共10兲is nodeless.兴It has to be mentioned that the solutions for M=0 and M=1 belong to different sextic potentials if sis the same, as the coefficient of the quadratic term is different then. We shall see, however, that with appropriate combinations of s and Mit is possible to solve sextic potentials that differ only in the strength of the centrifugal term. The normalization of the wave functions can also be given in closed form. For this one has to evaluate integrals of the type G. LÉVAI AND J. M. ARIAS PHYSICAL REVIEW C 69, 014304 (2004) 014304-2 I共A兲⬅ 冕 0 ⬁ xAexp 冉 −a 2x4−bx2 冊 =1 2⌫ 冉 A+1 2 冊 a−共A+1兲/4exp 冉 b2 4a 冊 D−共A+1兲/2 冉 b a1/2 冊 共11兲 =1 2⌫ 冉 A+1 2 冊 共2a兲共−A+1兲/4U 冉 A+1 4,1 2;b2 2a 冊 ,共12兲 where Dp共z兲is the parabolic cylinder function and U共 ␣ , ␤ ;z兲 is one of the forms of the confluent hypergeometric function 关14兴. Larger values of Mcan also be considered (e.g., for M =2 three different solutions are obtained for the three roots of a cubic algebraic equation for ␭), but M=0 and M=1 are sufficient for our purposes in this paper. A complete study including more solutions, explicit closed forms for the normalization factors, and applications to actual nuclei is underway. IV. APPLICATION AS A ␥ -INDEPENDENT POTENTIAL In order to cast Eq. (6)in a form similar to Eq. (5)we have to write x= ␤ and s=共 ␶ /2兲+5Ⲑ4(remember that ␶ 艌0). In order to keep the quadratic term at a constant value (once the a,bparameters are fixed)we also have to prescribe s+M+1 2=1 2 共 ␶ +2M+7 2 兲 ⬅c= const. 共13兲 With this the sextic oscillator Hamiltonian can be brought to the form of Eq. 共4兲with u共 ␤ 兲being u ␲ 共 ␤ 兲=共b2−4ac ␲ 兲 ␤ 2+2ab ␤ 4+a2 ␤ 6+u0 ␲ ,共14兲 where the index ␲ =± is included to distinguish the potential for even/odd ␶ ’s, which is slightly different as explained below. In Eq. 共14兲c ␲ are the constants obtained in Eq. 共13兲for even/odd values of ␶ and we have introduced a constant u0 ␲ for convenience, as will be discussed below. Equation (13)implies that increasing/decreasing Mby one unit has to come with decreasing/increasing ␶ by two units. Thus, once the values of the 共a,b兲parameters are fixed, the sequence of 共M, ␶ 兲values 共K,0兲,共K−1,2兲, 共K−2,4兲, ... correspond to solutions of Eq. (14)with c+=7 4 +K. In the same way, the sequence of 共M, ␶ 兲values 共K,1兲,共K−1,3兲,共K−2,5兲, ... correspond to solutions of Eq. (14)with c−=9 4+K. Consequently, the potential for ␶ -even and ␶ -odd states is slightly different due to the fact that the coupling coefficient b2−4ac±of the quadratic term is different in the two cases due to the choices for c+and c−that are necessary to separate the 共 ␶ +1兲共 ␶ +2兲 ␤ −2 term in a uniform way. This situation can be handled using different strategies. One possibility is setting u0 +=u0 −=0 and considering b2⬎10a, which minimizes the deviation of the quadratic terms compared to the quartic and sextic terms. Another possibility is introducing a relative energy shift between the ␶ -even and ␶ -odd potentials by setting u0 +and u0 −such that the potential minima are at the same energy. We shall discuss this possibility after analyzing qualitatively the spectrum and the potential shapes. Let us analyze the spectrum obtained in a simple case. Taking M=1 we obtain, as explained in the preceding section, two solutions for ␶ =0(one with no nodes, n=0, and the other one with one node, n=1). In the notation introduced in Ref. [8]the label ␰ is our n+1. Thus the two solutions of Eq. (10)with s=5/4 are ␾ n 共M=1兲共 ␤ 兲with n=0 and 1 and correspond to ␾ ␰ , ␶ = ␾ 1,0 and ␾ ␰ , ␶ = ␾ 2,0, respectively, in Ref. [8] notation. Inspecting the energy eigenvalues, the corresponding energies are E1共M=1, ␶ =0兲=E1,0=7b−2共b2+10a兲1/2 +u0 +and E2共M=1, ␶ =0兲=E2,0=7b+2共b2+10a兲1/2+u0 +, respectively. The same potential is obtained by taking M=0 and ␶ =2. In this case we have a single solution, Eq. (8)with s=9/4, ␾ 0 共M=0兲共 ␤ 兲with no nodes, and with energy E1共M =0, ␶ =2兲=9b+u0 +. This corresponds to ␾ ␰ , ␶ = ␾ 1,2 in the notation of Ref. [8]. A similar analysis can be performed for the solutions with odd- ␶ values. For M=1 there are two solutions with ␶ =1, Eq. (10)with s=7/4, which corresponds to ␾ ␰ , ␶ = ␾ 1,1 and ␾ ␰ , ␶ = ␾ 2,1 for n=0 and n=1, respectively. The corresponding energy eigenvalues are E1,1=9b−2共b2 +14a兲1/2+u0 −and E2,1=9b+2共b2+14a兲1/2+u0 −. Again the same potential is obtained for M=0 and ␶ =3. In this case there is a single solution with no nodes, Eq. (8)with s =11/4, which corresponds to ␾ 1,3 and has an energy E1,3 =11b+u0 −. In Fig. 1 a schematic spectrum is shown with indication of the relevant quantum numbers. In Fig. 2 the corresponding wave functions with the notation ␾ ␰ , ␶ are presented. Now we analyze the different potential shapes that can be produced by different election of parameters in Eq. (14). From Eq. (14)we find that the shape of the potential u ␲ 共 ␤ 兲 depends on the sign of b2−4ac ␲ and b, which sets the coefficients of the quadratic and quartic terms. (The coefficient of the leading sextic term is always positive.)When b2⬎4ac ␲ and b⬎0 hold [i.e., for b⬎2共ac ␲ 兲1/2], the potential has a minimum at ␤ =0 and it increases monotonously with ␤ . When b2⬍4ac ␲ , irrespective of the sign of b[i.e., for FIG. 1. Schematic typical spectrum for the sextic oscillator with indication of the relevant quantum numbers. SEXTIC OSCILLATOR AS A ␥ -INDEPENDENT POTENTIAL PHYSICAL REVIEW C 69, 014304 (2004) 014304-3 −2共ac ␲ 兲1/2⬍b⬍2共ac ␲ 兲1/2], a minimum appears for ␤ ⬎0, while for b2⬎4ac ␲ and b⬍0[i.e., for b⬍−2共ac ␲ 兲1/2], first a maximum appears and then a minimum as ␤ increases. In all three cases the exact location of the extremal point(s)can be obtained from the real and positive solutions of 共 ␤ 0 ␲ 兲2=1 3a关−2b±共b2+12ac ␲ 兲1/2兴.共15兲 Due to the relatively small difference in c+and c−, the ␶ -even and ␶ -odd potentials have the same types of extrema at about the same ␤ , except for some peculiar combinations of aand b. Assuming that there are no complications of this kind, we can now return to the question of renormalizing the minima of the ␶ -even and ␶ -odd potentials. For b⬎2共ac ␲ 兲1/2, ␲ = +, − the minima of the two potentials will be u0 +and u0 −at ␤ =0, so they coincide if u0 +=u0 −holds. For b⬍2共ac ␲ 兲1/2 we can equate the minima of u+共 ␤ 兲and u−共 ␤ 兲if we set u0 +=0 and u0 −=共b2−11a兲共 ␤ 0 +兲2−共b2−13a兲共 ␤ 0 −兲2+2ab关共 ␤ 0 +兲4−共 ␤ 0 −兲4兴 +a2关共 ␤ 0 +兲6−共 ␤ 0 −兲6兴,共16兲 where ␤ 0 ␲ are obtained from Eq. 共15兲with the choice of the “+” sign. With this the two potentials have their minima at the same energy, but they take on different values at the origin. Illustrations of the possible potential shapes are displayed in Fig. 3. Obviously, u0 −in Eq. 共15兲also has to be added to the energies of the ␶ -odd potential. Figure 4 shows the relative position of the energy levels E ␰ , ␶ as either aor b is varied and the other parameter is kept at a fixed value. The electric quadrupole transition rates can also be determined analytically by calculating the matrix elements of the transition operator [8,11] T共E2兲=t ␣ 2 ␮ =t ␤ 关D ␮ ,0 共2兲cos ␥ +2 −1/2共D ␮ ,2 共2兲+D ␮ ,−2 共2兲兲sin ␥ 兴. 共17兲 The radial integrals that appear in the ␤ variable in the matrix elements of T共E2兲can again be determined using Eq. 共11兲. FIG. 2. Wave functions with the notation ␾ ␰ , ␶ for the case of potential parameters a=40 000 and b=200. FIG. 3. Potentials u+共 ␤ 兲(full line)and u−共 ␤ 兲(broken line)for a=40 000, and b=1000 (left panel),b=200 (middle panel), and b =−1000 (right panel). The lowest energy level appears in these potentials at E1,0=4633.57, 73.35, and −9366.43, respectively. FIG. 4. Excitation energies E ␰ , ␶ *=E ␰ , ␶ −E1,0 with a=40 000 fixed as a function of b(left panel)and with b=200 fixed as a function of a(right panel). FIG. 5. The energy spectrum and the strength of some electric quadrupole transitions calculated with a=40 000 and b=200 (left panel)and the corresponding data for 134Ba (right panel). G. LÉVAI AND J. M. ARIAS PHYSICAL REVIEW C 69, 014304 (2004) 014304-4 In order to obtain the total matrix elements, one has to calculate also the components depending on ␥ and the Euler angles ␪ i. This can be done following the techniques described in Ref. 关16兴. These parts introduce certain selection rules not only for the angular momenta, but also for ␶ . V. DISCUSSION In order to compare the main characteristics of the sextic oscillator as a ␥ -unstable potential with those of other potentials of this kind, we present calculations for a particular value of the parameters, a=40 000 and b=200. These numbers were chosen such that the resulting energy spectrum approximates that of the 134Ba nucleus, the first candidate for E共5兲symmetry [17]. We stress that our aim is not to reproduce the experimental data, rather to get a qualitative picture about the general performance of the model. The potentials u±共 ␤ 兲are displayed in the middle panel of Fig. 3, while the energy eigenvalues are shown in Fig. 5, together with the corresponding experimental energy levels. Figure 5 also shows the calculated and the experimental B共E2兲values for transitions between the energy levels. Note that electric quadrupole transitions which change ␶ by more than one unit are zero if we use the transition operator (17), but finite B共E2兲strengths can be obtained if we apply terms of the next order (see, e.g., Ref. [18]). In Table I we summarize the ratio of the most important energy eigenvalues and those of the most characteristic B共E2兲transition rates obtained from the sextic oscillator with parameters a=40 000, b=200, the infinite square well potential [8], and the numerically solved ␤ 4potential [19]together with the corresponding experimental values for 134Ba, whenever available. It is seen that the energy ratios corresponding to the E共5兲symmetry systematically fall between the values of the ␤ 4potential and the sextic oscillator. The situation is less obvious for the ratio of the B共E2兲values: here the sextic oscillator and the infinite square well seem to yield similar ratios, while the numbers obtained from the ␤ 4potential are systematically higher. This might be due to the fact that the sextic oscillator potential goes to infinity steeper than the ␤ 4 potential, so the asymptotic behavior of its wave functions can be closer to that of the wave functions of the infinite square well. Comparing the results with the experimental data for 134Ba we can conclude that, at least in this case, the sextic oscillator allows a better approximation than the other essentially parameter-free potentials. We expect that this conclusion will be general due to the flexible nature of the sextic potential whose shape is governed by two parameters. In fact this potential can be used not just at the critical point but it can be useful to model the full shape phase transition from spherical to deformed ␥ -unstable nuclei by changing the parameters aand b. Before closing, we mention some aspects of the sextic oscillator that might give further help in the analysis of nuclei near critical points. First, we note that with M=2 in Eq. (6)the analysis can be extended to further states, such as ␾ 1,4, ␾ 1,5, ␾ 2,2, ␾ 2,3, ␾ 3,0, and ␾ 3,1. Second, the potential shape which contains both a local maximum and a minimum at ␤ ⬎0 might be useful in the description of nuclei with the so-called X共5兲symmetry, which is thought to occur in the shape phase transition between the spherical and the axially deformed domain [9]. Third, there are further quasi-exactly solvable potentials both with confining and nonconfining nature [15], which can also be considered in the Bohr Hamiltonian. ACKNOWLEDGMENTS This work was supported by the OTKA Grant No. T37502 (Hungary)and by the Spanish MCyT under Project No. BFM2002-03315. [1]A. Bohr and B. Mottelson, Nuclear Structure (Benjamin, Reading, MA, 1975), Vol. II. [2]F. Iachello and A. Arima, The Interacting Boson Mode (Cambridge University Press, Cambridge, 1987). [3]A. E. L. Dieperink, O. Scholten, and F. Iachello, Phys. Rev. Lett. 44, 1747 (1980). [4]D. H. Feng, R. Gilmore, and S. R. Deans, Phys. Rev. C 23, 1254 (1981). [5]E. López-Moreno and O. Castaños, Phys. Rev. C 54, 2374 (1996). [6]J. Jolie et al., Phys. Rev. Lett. 89, 182502 (2002). [7]J. M. Arias, J. Dukelsky, and J. E. García-Ramos, Phys. Rev. Lett. 91, 162502 (2003). [8]F. Iachello, Phys. Rev. Lett. 85, 3580 (2000). [9]F. Iachello, Phys. Rev. Lett. 87, 052502 (2001). [10]F. Iachello, Phys. Rev. Lett. 91, 132502 (2003). TABLE I. Ratios of some energy eigenvaluess and electric quadrupole transition strengths from the sextic oscillator with a=40 000, b =200, the infinite square well [8], and the ␤ 4potential [19], together with the experimentally observed quantities for 134Ba. E共41,2 +兲 E共21,1 +兲 E共02,0 +兲 E共21,1 +兲 E共61,3 +兲 E共21,1 +兲 B共E2;41,2 +→21,1 +兲 B共E2;21,1 +→01,0 +兲 B共E2;22,0 +→21,1 +兲 B共E2;21,1 +→01,0 +兲 B共E2;01,3 +→21,2 +兲 B共E2;21,1 +→01,0 +兲 Sextic oscillator 2.39 3.68 3.70 1.70 1.03 2.12 E共5兲2.20 3.03 3.59 1.68 0.86 2.21 ␤ 42.09 2.39 3.27 1.82 1.41 2.52 134Ba (expt.)2.31 3.57 3.65 1.56 (18)0.42 (12) SEXTIC OSCILLATOR AS A ␥ -INDEPENDENT POTENTIAL PHYSICAL REVIEW C 69, 014304 (2004) 014304-5 [11]L. Wilets and M. Jean, Phys. Rev. 102, 788 (1956). [12]J. P. Elliott, J. A. Evans, and P. Park, Phys. Lett. 169B, 309 (1986). [13]L. Fortunato and A. Vitturi, J. Phys. G 29, 1341 (2003). [14]M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970). [15]A. G. Ushveridze, Quasi-exactly Solvable Models in Quantum Mechanics (IOP, Bristol, 1994). [16]D. R. Bès, Nucl. Phys. 10, 379 (1959). [17]R. F. Casten and N. V. Zamfir, Phys. Rev. Lett. 85, 3584 (2000). [18]A. Frank, C. E. Alonso, and J. M. Arias, Phys. Rev. C 65, 014301 (2001). [19]J. M. Arias, C. E. Alonso, A. Vitturi, J. E. Garcia-Ramos, J. Dukelsky, and A. Frank, Phys. Rev. C 68, 041302(R)(2003). G. LÉVAI AND J. M. ARIAS PHYSICAL REVIEW C 69, 014304 (2004) 014304-6