Accurate calculation of resonances for a central-field model potential
aa r X i v : . [ m a t h - ph ] F e b Accurate calculation of resonances for acentral–field model potential
Francisco M. Fern´andez ‡ INIFTA (UNLP, CCT La Plata-CONICET), Divisi´on Qu´ımica Te´orica, Blvd. 113S/N, Sucursal 4, Casilla de Correo 16, 1900 La Plata, ArgentinaE-mail: [email protected] esonances for a central–field model Abstract.
We obtain accurate resonance energies for the Schr¨odinger equation witha central–field potential by means of a method based on a rational approximation tothe logarithmic derivative of the wavefunction. We discuss the rate of convergenceof our approach and compare present results with those obtained earlier by otherauthors. We show that present method is superior to the spherical–box approachapplied recently to the same problem. As far as we know present results are moreaccurate than those available in the literature and may be a suitable benchmark fortesting future approaches.
1. Introduction
In a recent paper Zhou et al [1] applied the well–known spherical–box stabilizationmethod to the calculation of the resonance energies of the Schr¨odinger equation withthe potential V ( r ) = V r e − r + Z/r . They integrated the eigenvalue equation by meansof the Runge–Kutta method and estimated the positions and widths of the resonancesfrom the behaviour of the bound–state energies as functions of the box radius. Inparticular they calculated the first two s–wave the first p–wave and the first d–waveresonances. Zhou et al [1] experienced some difficulties in estimating the position anwidth of the second s–wave resonance and could not obtain the third one. They alsoobtained rather crude estimations of the first p–wave and d–wave resonances. It seemsthat the spherical–box approach is rather ill–suited to broad resonances.The potential V ( r ) mentioned above has proved a suitable benchmark for thedevelopment and testing of several methods for the calculation of the energies ofmetastable states [2–24]. Most authors have considered the case Z = 0 [2–24] andjust a few ones included the Coulomb interaction Z = − V ( r ) with Z = 0 [17]. In that earlier paperwe did not discuss the rate of convergence of the method on this particular model andmerely showed the result for the lowest s–wave resonance. The purpose of this paper esonances for a central–field model Z = − V ( r )with Z = −
1, analyze its results and compare them with those obtained earlier by otherauthors. Finally, in Sec. 4 we discuss the advantages of the RPM and draw conclusions.
2. The method
The radial part of the dimensionless Schr¨odinger equation for a central–field potential V ( r ) is " − d dr + l ( l + 1)2 r + V ( r ) Φ( r ) = E Φ( r ) (1)where l = 0 , , . . . is the angular–momentum quantum number and Φ(0) = 0. The RPMis based on a rational approximation to the regularized logarithmic derivative of thewavefunction f ( r ) = l + 1 r − Φ ′ ( r )Φ( r ) (2)that can be expanded as follows: f ( r ) = ∞ X j =0 f j r j (3)Note that the term ( l + 1) /r removes the singularity of Φ ′ ( r ) / Φ( r ) at origin andthat we can obtain the coefficients f j ( E ) analytically by means of simple recurrencerelations [17].We then convert the Taylor series into a rational approximation:[ N + d/N ] = P N + dj =0 a j r j P Nj =0 b j r j = N + d +1 X j =0 f j r j (4)where N = 1 , , . . . and d = 0 , , . . . . The 2 N + d + 1 adjustable coefficients a j and b j areinsufficient to provide the 2 N + d +2 coefficients f j . This condition is satisfied only if theHankel determinant H dD ( E ) with matrix elements f i + j + d − ( E ), i, j = 1 , , . . . , D = N +1,vanishes [17] (and references therein). The RPM conjecture is that there are sequencesof roots E [ D,d ] of H dD ( E ) = 0, D = 2 , , . . . that converge towards the actual bound– and esonances for a central–field model
3. Results and discussion
For comparison we consider the potential V ( r ) = V r e − r + Zr (5)with the model parameters V = 7 . Z = − E l,ν so that Re E l,ν +1 > Re E l,ν , ν = 0 , , . . . . The s–,p– and d–waves discussed by Zhou et al [1] and Sofianos and Rakityansky [23] correspondto l = 0, l = 1, and l = 2, respectively.Since we are looking for Siegert pseudo states that satisfy [25]lim r →∞ Φ ′ ( r )Φ( r ) = ik (6)then it seems reasonable to choose d = 0 becauselim r →∞ [ N/N ] = a N b N (7)For that reason it should be assumed that d = 0 from now on, unless otherwise stated.In order to estimate the rate of convergence of the RPM we calculate L D =log | α D − α D +1 | where α D is either the real or imaginary part of E [ D, . Fig. 1 shows thatthe rate of convergence of the RPM for both the positions and widths of the first threes–wave resonances is remarkable. It is worth mentioning that in the case of a narrowresonance the imaginary part of the root will appear at sufficiently large determinantdimensions D ; that is to say, when | Re E [ D, − Re E [ D +1 , | is of the order of magnitudeof | Im E | . We appreciate that the rate of convergence (given approximately by theslope of L D ) is almost independent of ν ; the main difference is that the greater thevalue of ν the larger the determinant dimension D necessary for the appearance ofthe corresponding sequence. On the other hand, the performance of the spherical–boxapproach deteriorates as the resonance width increases [1]. It is clear that the RPM ispreferable to the spherical–box approach, at least for this example. esonances for a central–field model l = 1) is similar to that discussed above. It is clear that the rate of convergence of theHankel sequences is also independent of l . We confirm our conclusion that the RPM ispreferable to the spherical–box approach because Zhou et al [1] roughly estimated theposition and width of E , and were unable to obtain other p–wave resonances.Fig. 3 shows the rate of convergence for the first two d–wave resonances. Thebehaviour is similar to those discussed above for the s and p ones. According to Zhouet al [1] the spherical–box approach only revealed the first d resonance for which theycould provide a rather crude estimate of the position and width.As far as we know, the most accurate results for this model are those calculatedsome time ago by Sofianos and Rakityansky [23]. Present results are even more accurateand may therefore be a useful benchmark for other approaches. For that purpose weshow them in Table 1.Finally, we mention that the rate of convergence of the RPM is not affected bythe choice of the displacement d . In the present case, for example, we obtained similarresults with d = 1 that we do not show here.
4. Conclusions
One of the main advantages of the RPM is its remarkable simplicity. We first obtain thecoefficients of the Taylor series (3) by means of a straightforward recurrence relation [17].Second, we construct the Hankel determinant, which is a polynomial function of theenergy, and find its roots. Third, we identify the sequences of roots that convergetowards physically acceptable results.Another advantage of the RPM is that exactly the same Hankel determinant appliesto both the bound states and resonances. It comes from the fact that the RPM does nottake explicitly into account the asymptotic form of the wavefunction at infinity and therational approximation applies to any solution of the Schr¨odinger equation. Of course,we have to take into consideration the behaviour of the wavefunction at origin in order esonances for a central–field model ′ ( r ) / Φ( r ).We think that present results clearly show that the RPM is much more accurateand reliable than the spherical–box approximation. We have calculated the resonancesdiscussed by Zhou et al [1] with much more accuracy and also obtained others that thoseauthors were unable to locate. Besides, it is worth noting that the RPM is as simple,or even simpler, than the box–stabilization method in any of its forms [1, 6, 13, 14]. [1] Zhou S-G, Meng J, and Zhao E-G 2009 J. Phys. B J. Phys. B J. Phys. A L39.[4] Isaacson A D, McCurdy C W, and Miller W H 1978
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Some complex energies E l,ν for the potential (5) l ν Re E Im E = Γ /
20 0 1.7805245363623048 0.000047859698428760 1 4.101494946209 0.5786272137660 2 4.6634610967 2.68320077031 0 3.848001634811759 0.1376922295857681 1 4.750053489274 1.752789924361482 0 4.9005161468291143 0.78375350826658582 1 5.3006134902578 2.942357430621 DL D ν =0 ν =1 ν =2 Figure 1.
Convergence rate L D for the real (solid line) and imaginary (dashed line)parts of the energies E ,ν esonances for a central–field model DL D ν =0 ν =1 Figure 2.
Convergence rate L D for the real (solid line) and imaginary (dashed line)parts of the energies E ,ν DL D ν =0 ν =1 Figure 3.
Convergence rate L D for the real (solid line) and imaginary (dashed line)parts of the energies E2