Calculation of bound states and resonances in perturbed Coulomb models
aa r X i v : . [ m a t h - ph ] N ov Calculation of bound states and resonances inperturbed Coulomb models
Francisco M. Fern´andezINIFTA (Conicet, UNLP), Divisi´on Qu´ımica Te´orica,Diag. 113 y 64 (S/N), Sucursal 4, Casilla de Correo 16,1900 La Plata, ArgentinaE–mail: [email protected] 31, 2018
Abstract
We calculate accurate bound states and resonances of two interest-ing perturbed Coulomb models by means of the Riccati-Pad´e method.This approach is based on a rational approximation to a modifiedlogarithmic derivative of the eigenfunction and produces sequences ofroots of Hankel determinants that converge towards the eigenvaluesof the equation. Introduction
In a most interesting series of papers Killingeck et al [1–3] and Killingbeck[4, 5] have shown that perturbation theory and the Hill–series method aresuitable tools for the calculation of bound states and resonances of simplequantum–mechanical models. In order to obtain the complex eigenvaluesthat correspond to unstable states they resort to a complex parametrizationof the methods that they call “complexification ”.Another approach that proves useful for the accurate calculation of boundstates and resonances is the Riccati–Pad´e method (RPM) based on a rationalapproximation to a modified (or regularized) logarithmic derivative of theeigenfunction [6–15]. In this paper we apply the RPM to the interestingperturbed Coulomb problems discussed recently by Killingeck [5] with thepurpose of challenging the recently developed asymptotic iteration method[16–22].In Section 2 we outline the main features of the RPM. In Section 3 we dis-cuss a perturbed Coulomb model with interesting bound states. In Section 4we calculate the resonances for a slightly modified model with continuumstates. Finally, in Section 5 we draw conclusions on the performance of theRPM.
Suppose that we want to obtain sufficiently accurate solutions to the eigen-value equation ψ ′′ ( x ) + Q ( x ) ψ ( x ) = 0 , Q ( x ) = E − V ( x ) (1)2here Q ( x ) can be expanded as Q ( x ) = ∞ X j =0 Q j − x βj − (2)about x = 0. We transform the linear differential equation (1) into a Riccatione for the modified logarithmic derivative of the eigenfunction: f ( x ) = sx − ψ ′ ( x ) ψ ( x ) (3)On substituting (3) into (1) we obtain f ′ ( x ) + 2 sx f ( x ) − f ( x ) − Q ( x ) − s ( s − x = 0 (4)We choose s ( s −
1) = − Q − in order to remove the pole at origin, and, as aresult, the expansion f ( x ) = x β − ∞ X j =0 f j x βj (5)for the solution to the Riccati equation (4) converges in a neighbourhoodof x = 0. Notice that if we substitute the expansions (2) and (5) into theRiccati equation (4) we easily obtain the series coefficients f j as a functionof E and the known potential parameters Q j .We rewrite the partial sums of the expansion (5) as rational approxima-tions x β − [ N + d/N ]( z ), where z = x β , in such a way that[ N + d/N ]( z ) = P N + dj =0 a j z j P Nj =0 b j z j = N + d +1 X j =0 f j z j + O ( z N + d +2 ) (6)In order to satisfy this condition the Hankel determinant H dD , with matrixelements f i + j + d − , i, j = 1 , , . . . , D , vanishes. Here, D = N + 1 = 2 , , . . . is the determinant dimension and d = 0 , , . . . . The main assumption of theRiccati–Pad´e method (RPM) is that there is a sequence of roots E [ D,d ] of3 dD ( E ) = 0 for D = 2 , , . . . that converges towards a given eigenvalue ofequation (1). Comparison of sequences with different d values is useful toestimate the accuracy of the converged results.Notice that we do not have to take the boundary conditions explicitly intoaccount in order to apply the RPM; the approach selects them automatically.In addition to the answers expected from physical considerations, the RPMalso yields unwanted solutions as shown below. From the ansatz ϕ ( r, λ ) = r exp ( − r − λr ) and the equation ϕ ′′ ( r, λ ) / [2 ϕ ( r, λ )] = V ( r, λ ) − E ( λ ) we derive a potential–energy function V ( r, λ ) = − /r + 2 λr +2 λ r if E ( λ ) = − / λ . For λ > ϕ ( r, λ ) and E ( λ ) are a pair ofeigenfunction and eigenvalue of the Schr¨odinger equation with the potential V ( r, λ ). For λ < E ( λ ) = − / λ is not an eigenvalue of the Schr¨odingerequation because the corresponding eigenfunction ϕ ( r, λ ) is not square in-tegrable. Curiously enough, e ( λ ) = − / − λ is close to the ground–stateeigenvalue of the Schr¨odinger equation ψ ′′ ( r ) + 2 [ E − V ( r )] ψ ( r ) = 0 , V ( r ) = − r − λr + 2 λ r , λ > λ is sufficiently small. Killingbeck [5] calculated the energy shift ∆( λ ) = E ( λ ) − e ( λ ) very accurately for several values of λ by means of the Hill–seriesmethod.Our interest in this model stems from the fact that 1 /r − ϕ ′ ( r, − λ ) /ϕ ( r, − λ ) = 1 − λr is an exact rational function and therefore e ( λ ) will alwaysbe a root of the Hankel determinants even though it does not correspond to4 square–integrable eigenfunction if λ >
0. This unwanted solution willappear as an exact multiple root of the Hankel determinant, very close tothe physical one when λ is close to zero.If λ < /
27 the potential–energy function (7) exhibits three stationarypoints: a minimum at r <
0, a maximum at r ≥ r >
4. On the other hand, there is only a minimum at r < λ > /
27. Obviously, only the stationary points at r > r = 0will require many terms in order to take into account the shallow minimumthat will move away from origin as λ decreases. In this case we expect to facethe necessity of Hankel determinants of greater dimension in order to obtainthe shift to a given accuracy as λ decreases. This unfavourable situation isan interesting test for the RPM that has not been applied to this kind ofproblems before.The Hankel determinants are polynomial functions of ∆ and λ . For ex-ample, H D (∆ , λ ) = ∆ D − P D (∆ , λ ) and, therefore, the approximation to theenergy shift is given by a root of P D (∆ , λ ) = 0 that approaches the multipleroot ∆ = 0 as λ decreases. Table 1 shows ∆( λ ) for some values of λ calcu-lated with determinants of dimension D ≤
20. In order to estimate the laststable digit we compared the sequences of roots with d = 0 and d = 1. Asexpected from the argument above, the accuracy decreases as λ decreases ifwe do not increase the maximum value of D consistently, but in all cases wehave verified that there is a sequence of roots of the Hankel determinantsthat converge towards the ground–state eigenvalue. Present results agree5ith those calculated by Killingbeck by means of the Hill–series method [5]. It has already been shown that the RPM is a most efficient tool for thecalculation of resonances in the continuum of simple quantum–mechanicalmodels [11, 14, 15]. However, for completeness in what follows we considerthe potential–energy function V ( r ) = − r + 2 λr − λ r (8)that is closely related to the preceding one but does not support boundstates because it is unbounded from below as r → ∞ . In this case weexpect unstable or resonant states with complex eigenvalues that correspondto tunnelling from the Coulomb well.Table 2 shows present results obtained from Hankel sequences with D ≤
20. As in the preceding example we compared sequences with d = 0 and d = 1 in order to estimate the last stable digit. Our results agree with thosereported by Killingbeck [5], except for λ = 0 .
08. While the first digits of theimaginary part of our eigenvalue agree with those in Killingbeck’s Table 3 [5],the real part is completely different. The disagreement is due to a misprintin Killingbeck’s Table 3 for that particular entry. In fact, we have foundthat the real part of the eigenvalue reported by Killingbeck for λ = 0 . λ = 0 .
05 instead, as shown in present Table 2.6
Conclusions
We have shown that the RPM is suitable for the accurate calculation of boundstates and resonances of perturbed Coulomb problems. The first model,equation (7), considered in this paper exhibits interesting features that werenot faced in previous applications of the RMP [6–15]. A shallow minimumthat moves forward from origin as the potential parameter λ decreases makesit necessary to resort to Hankel determinants of increasing dimension in orderto obtain eigenvalues of a given accuracy. On the other hand, we had alreadyproved that the RPM is suitable for the calculation of resonances in thecontinuum, and we simply confirmed this strength of the approach by meansof the second model chosen above.The applicability of the RPM is not restricted to eigenvalue equations. Wehave recently applied it to several ordinary nonlinear differential equations[23]. Since most of them are not Riccati equations we called this variant ofthe method Pad´e–Hankel, but the strategy is basically the same outlined inthis paper. References [1] J. P. Killingbeck, A. Grosjean, and G. Jolicard, J. Phys. A 38 (2005)L695-L699.[2] J. P. Killingbeck, A. Grosjean, and G. Jolicard, J. Phys. A 39 (2006)L547-L550.[3] J. P. Killingbeck, A. Grosjean, and G. Jolicard, J. Phys. A 37 (2007)74] J. P. Killinbeck, J. Phys. A 40 (2007) 9017-9024.[5] J. P. Killingbeck, J. Phys. A 40 (2007) 2819-2824.[6] F. M. Fern´andez, Q. Ma, and R. H. Tipping, Phys. Rev. A 40 (1989)6149-6153.[7] F. M. Fern´andez, Q. Ma, and R. H. Tipping, Phys. Rev. A 39 (1989)1605-1609.[8] F. M. Fern´andez, Phys. Lett. A 166 (1992) 173-176.[9] F. M. Fern´andez and R. Guardiola, J. Phys. A 26 (1993) 7169-7180.[10] F. M. Fern´andez, Phys. Lett. A 203 (1995) 275-278.[11] F. M. Fern´andez, J. Phys. A 28 (1995) 4043-4051.[12] F. M. Fern´andez, J. Chem. Phys. 103 (1995) 6581-6585.[13] F. M. Fern´andez, J. Phys. A 29 (1996) 3167-3177.[14] F. M. Fern´andez, Phys. Rev. A 54 (1996) 1206-1209.[15] F. M. Fern´andez, Chem. Phys. Lett 281 (1997) 337-342.[16] H. Ciftci, R. L. Hall, and N. Saad, J. Phys. A 36 (2003) 11807-11816.[17] F. M. Fern´andez, J. Phys. A 37 (2004) 6173-6180.[18] T. Barakat, K. Abodayeh, and A. Mukheimer, J. Phys. A 38 (2005)1299-1304.[19] T. Barakat, Phys. Lett. A 344 (2005) 411-417.820] H. Ciftci, R. L. Hall, and N. Saad, Phys. Lett. A 340 (2005) 388-396.[21] H. Ciftci, R. L. Hall, and N. Saad, J. Phys. A 38 (2005) 1147-1155.[22] F. M. Fern´andez, Phys. Lett. A 346 (2005) 381-383.[23] P. Amore and F. M. Fern´andez, Rational Approximation for Two-PointBoundary value problems, arXiv:0705.38629able 1: Energy shift ∆( λ ) for the ground–state energy of the perturbedCoulomb model (7) λ ∆( λ )0.10 3 . − . − . − . − . − . − . − Table 2: Resonance for the 1 s state of the perturbed Coulomb model (8) λ Re E Im E − . . − − . . − − . . − − . − − −− − −