We obtain accurate eigenvalues for two recently derived SUSY partner
Hamiltonians. We improve the Rayleigh-Ritz variational method proposed by the
authors and show how to apply the Riccati-Pad\'{e} method to those particular
partner potentials

Fernández, Francisco M.

Open Physics

English

arXiv

We obtain accurate eigenvalues for two recently derived SUSY partner Hamiltonians. We improve the Rayleigh-Ritz variational method proposed by the authors and show how to apply the Riccati-Padé method to those particular partner potentials.Facultad de Ciencias ExactasInstituto de Investigaciones Fisicoquímicas Teóricas y Aplicada

Fernández, Francisco Marcelo

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

Cent. Eur. J. Phys. • 11(4) • 2013 • 470-473DOI: 10.2478/s11534-013-0198-0Central European Journal of PhysicsAccurate calculation of the eigenvalues of a newsimple class of superpotentials in SUSY quantummechanicsResearch ArticleFrancisco M. Fernández∗INIFTA (UNLP, CCT La Plata-CONICET), División Química Teórica,Blvd. 113 S/N, Sucursal 4, Casilla de Correo 16,1900 La Plata, ArgentinaReceived 04 July 2012; accepted 17 February 2013Abstract: We obtain accurate eigenvalues for two recently derived SUSY partner Hamiltonians. We improve theRayleigh-Ritz variational method proposed by the authors and show how to apply the Riccati-Padé methodto those particular partner potentials.PACS (2008): 03.65.GeKeywords: SUSY partner Hamiltonians • variational calculation • Riccati-Padé method • accurate eigenvalues© Versita sp. z o.o.1. IntroductionIn a recent paper Marques et al. [1] derived two new simplesupersymmetric partner potentials and showed that theground-state eigenfunction of one of the Hamiltonian op-erators can be obtained exactly. In order to calculate theremaining states they resorted to the Rayleigh-Ritz vari-ational method. However, the convergence rate of theirapproach appears to be insufficient to reveal the rela-tion between the eigenvalues of the partner Hamiltoniansclearly.The authors also estimated the ground-state energy of thequartic anharmonic oscillator by means of perturbation∗E-mail: fernande@quimica.unlp.edu.artheory starting from the known solution for one of thepartner Hamiltonian operators.The purpose of this paper is twofold. First we improve thevariational method proposed by Marques et al. [1] and,second, we show that the Riccati-Padé method (RPM) [2–4] provides highly accurate results for the eigenvalues ofthose partner Hamiltonian operators. In addition, we showthat exactly the same Rayleigh-Ritz variational methodapplied to the SUSY partner Hamiltonians is also suit-able for the calculation of the eigenvalues of the quarticanharmonic oscillator.In Section 2 we apply the Rayleigh-Ritz variationalmethod and the RPM to the pair of SUSY Hamiltoni-ans proposed by Marques et al. [1] and also to the quarticoscillator. In Section 3 we comment on those approachesand draw conclusions.470UnauthenticatedDownload Date | 10/28/19 9:48 PMFrancisco M. Fernández2. SUSY partnersTable 1. Eigenvalues E−n from the Rayleigh-Ritz method.N n = 0 n = 1 n = 2 n = 3 n = 42 0 1.970246841 5.765408776 9.4885425543 0 1.970246137 5.511879703 9.488418664 14.130078374 0 1.969640134 5.507908922 9.406978612 13.892661955 0 1.969515816 5.507440820 9.394523525 13.874350606 0 1.969507628 5.507202146 9.394324659 13.859572597 0 1.969507538 5.507178381 9.394287127 13.85838650n = 5 n = 6 n = 7 n = 823 19.489629264 19.02107688 24.09989393 33.111071725 18.66009505 24.07406802 29.81927599 35.912983876 18.64895855 23.84777162 29.26028194 35.656564487 18.64704858 23.80790714 29.25748454 34.98577701Table 2. Eigenvalues E+n from the Rayleigh-Ritz method.N n = 0 n = 1 n = 2 n = 32 1.970246841 5.529040685 9.488542554 14.357212103 1.970246137 5.514418950 9.488418664 14.001035344 1.969640134 5.510107538 9.406978612 13.925448755 1.969515816 5.507493533 9.394523525 13.863892186 1.969507628 5.507185747 9.394324659 13.858512527 1.969507538 5.507178915 9.394287127 13.85851126n = 4 n = 5 n = 6 n = 723 19.48962926 24.528380464 19.02107688 24.52809061 33.11107172 37.633352295 18.66009505 23.91353748 29.81927599 36.331674416 18.64895855 23.81074205 29.26028194 35.025772447 18.64704858 23.81074194 29.25748454 34.98424967The partner Hamiltonian operators are given by [1]:H± = − d2dx2 + x4 ± 2|x|, (1)where we have chosen g = 1 without loss of generality.In order to solve the Schrödinger equationH±ψ±n = E±n ψ±n (2)approximately, Marques et al. [1] chose the nonorthogonalbasis setφj (x) = x j−1e−|x|3/3, j = 1, 2, . . . (3)Table 3. Eigenvalues for the partner SUSY Hamiltonian operatorsfrom the Riccati-Padé method.Even (s = 0) Odd (s = 1)0 1.96950751375029482491.9695075137502948249 5.50717777714596996765.5071777771459699676 9.39426743787389146589.3942674378738914658 13.85837193654130014713.858371936541300147 18.64597563398844479918.645975633988444799 23.80718591798576691823.8071859179857669 29.232554550621496129.23255455062149608 34.946335718890610634.94633571889061and derived a single secular equation for both the evenand odd states. Since states of different parity do not mixthe dimension of their secular equation is twice the sizeof what would be actually necessary for a calculation ofthe same accuracy.In this paper we propose to work on the half line x ∈[0,∞) taking into account the appropriate boundary con-ditions at the origin: ψ(0) 6= 0, ψ ′(0) = 0 for the evenstates and ψ(0) = 0, ψ ′(0) 6= 0 for the odd ones. Theappropriate basis sets areφej = x je−x3/3, j = 0, 2, 3, . . . ,φoj = x je−x3/3, j = 1, 2, 3, . . . , (4)for the former and latter case, respectively.Tables 1 and 2 show Rayleigh-Ritz results for the firsteigenvalues of H− and H+, respectively, in terms of thesize N of the basis sets. We appreciate that the varia-tional results decrease with the number of basis functionsas expected. However, we are not aware of any proof thateither of the basis sets (3) or (4) is complete. We see thatpresent results are smaller, and therefore more accurate,than those of Marques et al. [1]. In addition, the resultsin their tables 1 and 2 exhibit unchanged entries whenthe size of the basis set increases from m to m+ 1 whichreflects the fact that the added basis function improvesthe even (odd) state but has no effect on the odd (even)one. As a result their matrices are twice the size of whatis necessary for a calculation of the same accuracy.The results of Tables 1 and 2, as well as those of Mar-ques et al. [1], suggest that E−n = E+n−1, n = 1, 2, . . ., asexpected for a SUSY pair of partner Hamiltonian opera-tors. However, those variational results are not sufficientlyaccurate and, as argued above, we are not aware of anyproof of convergence. In order to obtain more accurate re-sults we resort to the RPM [2–4]. To this end we consider471UnauthenticatedDownload Date | 10/28/19 9:48 PMAccurate calculation of the eigenvalues of a new simple class of superpotentials in SUSY quantum mechanicsTable 4. Eigenvalues En for the quartic oscillator.N n = 0 n = 1 n = 22 1.077335422 3.804924324 8.1025312123 1.061889825 3.802939362 7.4864878664 1.060417725 3.800572274 7.4567380185 1.060362727 3.799746874 7.4565408806 1.060362727 3.799674368 7.4558264967 1.060362223 3.799673299 7.455703721RPM 1.0603620904841828996 3.7996730298013941688 7.4556979379867383922n = 3 n = 4 n = 52 11.867596773 11.77079816 16.91718138 21.922326894 11.67894913 16.29307103 21.798834885 11.64634787 16.29029658 21.285039336 11.64483615 16.26577116 21.241385537 11.64480669 16.26191320 21.24054440RPM 11.644745511378162021 16.261826018850225938 21.238372918235940024the modified logarithmic derivative of the wavefunction ψf (x) = sx − ψ ′(x)ψ(x) , (5)where s = 0 or s = 1 for even or odd states, respectively.It satisfies the Riccati equationf ′(x) = f (x)2 − 2sx f (x) + E − x4 ∓ 2x, x > 0 (6)and can be expanded in a Taylor series about the originf (x) = ∞∑j=0 fjxj , (7)where the coefficients fj can be easily obtained from theRiccati equation (6). We have f0 = −ψ ′(0)/ψ(0) = 0 forthe even states and choose f0 = 0 for the odd ones in orderto remove the unbalanced term 2f0/x that appears whenwe substitute the expansion (7) into the Riccati equation(6). Note that this application of the RPM to even po-tentials is different from the one in earlier papers [2–4]because in the present case we restrict the calculation tothe half line. The reason for such a modification is thatthe present even potentials cannot be expanded in an x2-power series.With the coefficients of the expansion (7) we construct theHankel determinants HdD = ∣∣fj+j+d−1∣∣Di,j=1 that are polyno-mial functions of the energy and obtain the approximateeigenvalues from the roots of HdD(E) = 0 [2–4]. More pre-cisely, we expect that sequences of roots E [D,d]n convergetowards the eigenvalues En, n = 0, 1, . . ., of the corre-sponding Hamiltonian operator as D increases. Since theSUSY partner potentials on the half line V±(x) = x4 ± 2xsatisfy V+(−x) = V−(x) then the RPM yields the eigen-values of both partner Hamiltonians simultaneously [5].More precisely, the sequences of roots for the even andodd wavefunctions will converge towards E−0 < E+0 <E−2 < E+2 < . . . and E−1 < E+1 < E−3 < E+3 < . . ., respec-tively.Table 3 shows the lowest eigenvalues for the SUSY part-ner Hamiltonians estimated from Hankel sequences withD ≤ 30, d = 0 and d = 1. The RPM yields the ground-state eigenvalue of H− exactly (E−0 = 0) and there is nodoubt from present accurate eigenvalues that E−n = E+n−1.Marques et al. [1] estimated the ground-state eigenvaluesfor the Schrödinger equation with the potentials V (x) = x4and V+(x) from the exact ground-state solution ψ−0 , E−0 forH−. They resorted to logarithmic perturbation theory buttheir results were not accurate. Here, we point out thatthey could have obtained more accurate results by meansof the variational method discussed above. In Table 4 weshow the results coming from the more efficient Rayleigh-Ritz variational method proposed in this paper and wealso add the accurate eigenvalues produced by the RPMfor comparison.3. ConclusionsIt is well known that the Rayleigh-Ritz variational methodis more efficient if one treats every symmetry species sepa-rately. In the present case we have carried out the separa-tion into even and odd functions on the half line by means472UnauthenticatedDownload Date | 10/28/19 9:48 PMFrancisco M. Fernándezof appropriate boundary conditions at origin. Thus, thesize of present secular equations, and consequently thecomputational effort for a given accuracy, is one half thatof Marques et al. [1].Working on the half line and taking into account theboundary conditions at origin also enables one to obtainhighly accurate eigenvalues by means of the RPM. In thepresent case one has to be cautious because the sameHankel determinant exhibits roots close to the eigenval-ues of both partner Hamiltonian operators. However, itis not difficult to identify the sequence of roots that con-verges to a chosen eigenvalue. The present RPM resultsconfirm that the two Hamiltonian operators proposed byMarques et al. [1] are already SUSY partners.The RPM was applied to sequences of SUSY partner po-tentials in the past [6]. However, the main interest in thatpaper was to obtain excited-state eigenvalues with Han-kel determinants of relatively small dimension. Besides,those potentials can be expanded in x2-power series andcan therefore be treated in the usual way.References[1] F. Marques, O. Negrini, A. J. da Silva, J. Phys. A 45,115307 (2012)[2] F. M. Fernández, Q. Ma, R. H. Tipping, Phys. Rev. A39, 1605 (1989)[3] F. M. Fernández, Q. Ma, R. H. Tipping, Phys. Rev. A40, 6149 (1989)[4] F. M. Fernández, arXiv:0807.0655v2 [math-ph][5] F. M. Fernández, J. Phys. A 29, 3167 (1996)[6] F. M. Fernández, Q. Ma, D. J. DeSmet, R. H. Tipping,Can. J. Phys. 67, 931 (1989)473UnauthenticatedDownload Date | 10/28/19 9:48 PM

Accurate calculation of the eigenvalues of a new simple class of superpotentials in SUSY quantum mechanics

Abstract
               We obtain accurate eigenvalues for two recently derived SUSY partner Hamiltonians. We improve the Rayleigh-Ritz variational method proposed by the authors and show how to apply the Riccati-Padé method to those particular partner potentials.</jats:p

Francisco Fernández

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Cent. Eur. J. Phys. • 11(4) • 2013 • 470-473
DOI: 10.2478/s11534-013-0198-0
Central European Journal of Physics
Accurate calculation of the eigenvalues of a new
simple class of superpotentials in SUSY quantum
mechanics
Research Article
Francisco M. Fernández∗
INIFTA (UNLP, CCT La Plata-CONICET), División Química Teórica,
Blvd. 113 S/N, Sucursal 4, Casilla de Correo 16,
1900 La Plata, Argentina
Received 04 July 2012; accepted 17 February 2013
Abstract: We obtain accurate eigenvalues for two recently derived SUSY partner Hamiltonians. We improve the
Rayleigh-Ritz variational method proposed by the authors and show how to apply the Riccati-Padé method
to those particular partner potentials.
PACS (2008): 03.65.Ge
Keywords: SUSY partner Hamiltonians • variational calculation • Riccati-Padé method • accurate eigenvalues
© Versita sp. z o.o.
1. Introduction
In a recent paper Marques et al. [1] derived two new simplesupersymmetric partner potentials and showed that theground-state eigenfunction of one of the Hamiltonian op-erators can be obtained exactly. In order to calculate theremaining states they resorted to the Rayleigh-Ritz vari-ational method. However, the convergence rate of theirapproach appears to be insufficient to reveal the rela-tion between the eigenvalues of the partner Hamiltoniansclearly.
The authors also estimated the ground-state energy of thequartic anharmonic oscillator by means of perturbation
∗E-mail: fernande@quimica.unlp.edu.ar
theory starting from the known solution for one of thepartner Hamiltonian operators.
The purpose of this paper is twofold. First we improve thevariational method proposed by Marques et al. [1] and,second, we show that the Riccati-Padé method (RPM) [2–4] provides highly accurate results for the eigenvalues ofthose partner Hamiltonian operators. In addition, we showthat exactly the same Rayleigh-Ritz variational methodapplied to the SUSY partner Hamiltonians is also suit-able for the calculation of the eigenvalues of the quarticanharmonic oscillator.
In Section 2 we apply the Rayleigh-Ritz variationalmethod and the RPM to the pair of SUSY Hamiltoni-ans proposed by Marques et al. [1] and also to the quarticoscillator. In Section 3 we comment on those approachesand draw conclusions.
470
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Francisco M. Fernández
2. SUSY partners
Table 1. Eigenvalues E−n from the Rayleigh-Ritz method.
N n = 0 n = 1 n = 2 n = 3 n = 42 0 1.970246841 5.765408776 9.4885425543 0 1.970246137 5.511879703 9.488418664 14.130078374 0 1.969640134 5.507908922 9.406978612 13.892661955 0 1.969515816 5.507440820 9.394523525 13.874350606 0 1.969507628 5.507202146 9.394324659 13.859572597 0 1.969507538 5.507178381 9.394287127 13.85838650n = 5 n = 6 n = 7 n = 823 19.489629264 19.02107688 24.09989393 33.111071725 18.66009505 24.07406802 29.81927599 35.912983876 18.64895855 23.84777162 29.26028194 35.656564487 18.64704858 23.80790714 29.25748454 34.98577701
Table 2. Eigenvalues E+n from the Rayleigh-Ritz method.
N n = 0 n = 1 n = 2 n = 32 1.970246841 5.529040685 9.488542554 14.357212103 1.970246137 5.514418950 9.488418664 14.001035344 1.969640134 5.510107538 9.406978612 13.925448755 1.969515816 5.507493533 9.394523525 13.863892186 1.969507628 5.507185747 9.394324659 13.858512527 1.969507538 5.507178915 9.394287127 13.85851126n = 4 n = 5 n = 6 n = 723 19.48962926 24.528380464 19.02107688 24.52809061 33.11107172 37.633352295 18.66009505 23.91353748 29.81927599 36.331674416 18.64895855 23.81074205 29.26028194 35.025772447 18.64704858 23.81074194 29.25748454 34.98424967
The partner Hamiltonian operators are given by [1]:
H± = − d2dx2 + x4 ± 2|x|, (1)
where we have chosen g = 1 without loss of generality.In order to solve the Schrödinger equation
H±ψ±n = E±n ψ±n (2)
approximately, Marques et al. [1] chose the nonorthogonalbasis set
φj (x) = x j−1e−|x|3/3, j = 1, 2, . . . (3)
Table 3. Eigenvalues for the partner SUSY Hamiltonian operators
from the Riccati-Padé method.
Even (s = 0) Odd (s = 1)0 1.96950751375029482491.9695075137502948249 5.50717777714596996765.5071777771459699676 9.39426743787389146589.3942674378738914658 13.85837193654130014713.858371936541300147 18.64597563398844479918.645975633988444799 23.80718591798576691823.8071859179857669 29.232554550621496129.23255455062149608 34.946335718890610634.94633571889061
and derived a single secular equation for both the evenand odd states. Since states of different parity do not mixthe dimension of their secular equation is twice the sizeof what would be actually necessary for a calculation ofthe same accuracy.In this paper we propose to work on the half line x ∈[0,∞) taking into account the appropriate boundary con-ditions at the origin: ψ(0) 6= 0, ψ ′(0) = 0 for the evenstates and ψ(0) = 0, ψ ′(0) 6= 0 for the odd ones. Theappropriate basis sets are
φej = x je−x3/3, j = 0, 2, 3, . . . ,φoj = x je−x3/3, j = 1, 2, 3, . . . , (4)
for the former and latter case, respectively.Tables 1 and 2 show Rayleigh-Ritz results for the firsteigenvalues of H− and H+, respectively, in terms of thesize N of the basis sets. We appreciate that the varia-tional results decrease with the number of basis functionsas expected. However, we are not aware of any proof thateither of the basis sets (3) or (4) is complete. We see thatpresent results are smaller, and therefore more accurate,than those of Marques et al. [1]. In addition, the resultsin their tables 1 and 2 exhibit unchanged entries whenthe size of the basis set increases from m to m+ 1 whichreflects the fact that the added basis function improvesthe even (odd) state but has no effect on the odd (even)one. As a result their matrices are twice the size of whatis necessary for a calculation of the same accuracy.The results of Tables 1 and 2, as well as those of Mar-ques et al. [1], suggest that E−n = E+n−1, n = 1, 2, . . ., asexpected for a SUSY pair of partner Hamiltonian opera-tors. However, those variational results are not sufficientlyaccurate and, as argued above, we are not aware of anyproof of convergence. In order to obtain more accurate re-sults we resort to the RPM [2–4]. To this end we consider
471
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Accurate calculation of the eigenvalues of a new simple class of superpotentials in SUSY quantum mechanics
Table 4. Eigenvalues En for the quartic oscillator.
N n = 0 n = 1 n = 22 1.077335422 3.804924324 8.1025312123 1.061889825 3.802939362 7.4864878664 1.060417725 3.800572274 7.4567380185 1.060362727 3.799746874 7.4565408806 1.060362727 3.799674368 7.4558264967 1.060362223 3.799673299 7.455703721RPM 1.0603620904841828996 3.7996730298013941688 7.4556979379867383922n = 3 n = 4 n = 52 11.867596773 11.77079816 16.91718138 21.922326894 11.67894913 16.29307103 21.798834885 11.64634787 16.29029658 21.285039336 11.64483615 16.26577116 21.241385537 11.64480669 16.26191320 21.24054440RPM 11.644745511378162021 16.261826018850225938 21.238372918235940024
the modified logarithmic derivative of the wavefunction ψ
f (x) = sx − ψ ′(x)ψ(x) , (5)
where s = 0 or s = 1 for even or odd states, respectively.It satisfies the Riccati equation
f ′(x) = f (x)2 − 2sx f (x) + E − x4 ∓ 2x, x > 0 (6)
and can be expanded in a Taylor series about the origin
f (x) = ∞∑j=0 fjx j , (7)
where the coefficients fj can be easily obtained from theRiccati equation (6). We have f0 = −ψ ′(0)/ψ(0) = 0 forthe even states and choose f0 = 0 for the odd ones in orderto remove the unbalanced term 2f0/x that appears whenwe substitute the expansion (7) into the Riccati equation(6). Note that this application of the RPM to even po-tentials is different from the one in earlier papers [2–4]because in the present case we restrict the calculation tothe half line. The reason for such a modification is thatthe present even potentials cannot be expanded in an x2-power series.With the coefficients of the expansion (7) we construct theHankel determinants HdD = ∣∣fj+j+d−1∣∣Di,j=1 that are polyno-mial functions of the energy and obtain the approximateeigenvalues from the roots of HdD(E) = 0 [2–4]. More pre-cisely, we expect that sequences of roots E [D,d]n converge
towards the eigenvalues En, n = 0, 1, . . ., of the corre-sponding Hamiltonian operator as D increases. Since theSUSY partner potentials on the half line V±(x) = x4 ± 2xsatisfy V+(−x) = V−(x) then the RPM yields the eigen-values of both partner Hamiltonians simultaneously [5].More precisely, the sequences of roots for the even andodd wavefunctions will converge towards E−0 < E+0 <E−2 < E+2 < . . . and E−1 < E+1 < E−3 < E+3 < . . ., respec-tively.Table 3 shows the lowest eigenvalues for the SUSY part-ner Hamiltonians estimated from Hankel sequences withD ≤ 30, d = 0 and d = 1. The RPM yields the ground-state eigenvalue of H− exactly (E−0 = 0) and there is nodoubt from present accurate eigenvalues that E−n = E+n−1.Marques et al. [1] estimated the ground-state eigenvaluesfor the Schrödinger equation with the potentials V (x) = x4and V+(x) from the exact ground-state solution ψ−0 , E−0 forH−. They resorted to logarithmic perturbation theory buttheir results were not accurate. Here, we point out thatthey could have obtained more accurate results by meansof the variational method discussed above. In Table 4 weshow the results coming from the more efficient Rayleigh-Ritz variational method proposed in this paper and wealso add the accurate eigenvalues produced by the RPMfor comparison.
3. Conclusions
It is well known that the Rayleigh-Ritz variational methodis more efficient if one treats every symmetry species sepa-rately. In the present case we have carried out the separa-tion into even and odd functions on the half line by means
472
Unauthenticated
Download Date | 10/28/19 9:48 PM
Francisco M. Fernández
of appropriate boundary conditions at origin. Thus, thesize of present secular equations, and consequently thecomputational effort for a given accuracy, is one half thatof Marques et al. [1].Working on the half line and taking into account theboundary conditions at origin also enables one to obtainhighly accurate eigenvalues by means of the RPM. In thepresent case one has to be cautious because the sameHankel determinant exhibits roots close to the eigenval-ues of both partner Hamiltonian operators. However, itis not difficult to identify the sequence of roots that con-verges to a chosen eigenvalue. The present RPM resultsconfirm that the two Hamiltonian operators proposed byMarques et al. [1] are already SUSY partners.The RPM was applied to sequences of SUSY partner po-tentials in the past [6]. However, the main interest in thatpaper was to obtain excited-state eigenvalues with Han-kel determinants of relatively small dimension. Besides,
those potentials can be expanded in x2-power series andcan therefore be treated in the usual way.
References
[1] F. Marques, O. Negrini, A. J. da Silva, J. Phys. A 45,115307 (2012)[2] F. M. Fernández, Q. Ma, R. H. Tipping, Phys. Rev. A39, 1605 (1989)[3] F. M. Fernández, Q. Ma, R. H. Tipping, Phys. Rev. A40, 6149 (1989)[4] F. M. Fernández, arXiv:0807.0655v2 [math-ph][5] F. M. Fernández, J. Phys. A 29, 3167 (1996)[6] F. M. Fernández, Q. Ma, D. J. DeSmet, R. H. Tipping,Can. J. Phys. 67, 931 (1989)
473
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Central European Journal of Physics

We obtain accurate eigenvalues for two recently derived SUSY partner Hamiltonians. We improve the Rayleigh-Ritz variational method proposed by the authors and show how to apply the Riccati-Padé method to those particular partner potentialsFil: Fernández, Francisco Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico la Plata. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas; Argentina. Universidad Nacional de La Plata; Argentin

Cent. Eur. J. Phys. • 11(4) • 2013 • 470-473DOI: 10.2478/s11534-013-0198-0Central European Journal of PhysicsAccurate calculation of the eigenvalues of a newsimple class of superpotentials in SUSY quantummechanicsResearch ArticleFrancisco M. Fernández∗INIFTA (UNLP, CCT La Plata-CONICET), División Química Teórica,Blvd. 113 S/N, Sucursal 4, Casilla de Correo 16,1900 La Plata, ArgentinaReceived 04 July 2012; accepted 17 February 2013Abstract: We obtain accurate eigenvalues for two recently derived SUSY partner Hamiltonians. We improve theRayleigh-Ritz variational method proposed by the authors and show how to apply the Riccati-Padé methodto those particular partner potentials.PACS (2008): 03.65.GeKeywords: SUSY partner Hamiltonians • variational calculation • Riccati-Padé method • accurate eigenvalues© Versita sp. z o.o.1. IntroductionIn a recent paper Marques et al. [1] derived two new simplesupersymmetric partner potentials and showed that theground-state eigenfunction of one of the Hamiltonian op-erators can be obtained exactly. In order to calculate theremaining states they resorted to the Rayleigh-Ritz vari-ational method. However, the convergence rate of theirapproach appears to be insufficient to reveal the rela-tion between the eigenvalues of the partner Hamiltoniansclearly.The authors also estimated the ground-state energy of thequartic anharmonic oscillator by means of perturbation∗E-mail: fernande@quimica.unlp.edu.artheory starting from the known solution for one of thepartner Hamiltonian operators.The purpose of this paper is twofold. First we improve thevariational method proposed by Marques et al. [1] and,second, we show that the Riccati-Padé method (RPM) [2–4] provides highly accurate results for the eigenvalues ofthose partner Hamiltonian operators. In addition, we showthat exactly the same Rayleigh-Ritz variational methodapplied to the SUSY partner Hamiltonians is also suit-able for the calculation of the eigenvalues of the quarticanharmonic oscillator.In Section 2 we apply the Rayleigh-Ritz variationalmethod and the RPM to the pair of SUSY Hamiltoni-ans proposed by Marques et al. [1] and also to the quarticoscillator. In Section 3 we comment on those approachesand draw conclusions.470Francisco M. Fernández2. SUSY partnersTable 1. Eigenvalues E−n from the Rayleigh-Ritz method.N n = 0 n = 1 n = 2 n = 3 n = 42 0 1.970246841 5.765408776 9.4885425543 0 1.970246137 5.511879703 9.488418664 14.130078374 0 1.969640134 5.507908922 9.406978612 13.892661955 0 1.969515816 5.507440820 9.394523525 13.874350606 0 1.969507628 5.507202146 9.394324659 13.859572597 0 1.969507538 5.507178381 9.394287127 13.85838650n = 5 n = 6 n = 7 n = 823 19.489629264 19.02107688 24.09989393 33.111071725 18.66009505 24.07406802 29.81927599 35.912983876 18.64895855 23.84777162 29.26028194 35.656564487 18.64704858 23.80790714 29.25748454 34.98577701Table 2. Eigenvalues E+n from the Rayleigh-Ritz method.N n = 0 n = 1 n = 2 n = 32 1.970246841 5.529040685 9.488542554 14.357212103 1.970246137 5.514418950 9.488418664 14.001035344 1.969640134 5.510107538 9.406978612 13.925448755 1.969515816 5.507493533 9.394523525 13.863892186 1.969507628 5.507185747 9.394324659 13.858512527 1.969507538 5.507178915 9.394287127 13.85851126n = 4 n = 5 n = 6 n = 723 19.48962926 24.528380464 19.02107688 24.52809061 33.11107172 37.633352295 18.66009505 23.91353748 29.81927599 36.331674416 18.64895855 23.81074205 29.26028194 35.025772447 18.64704858 23.81074194 29.25748454 34.98424967The partner Hamiltonian operators are given by [1]:H± = − d2dx2 + x4 ± 2|x|, (1)where we have chosen g = 1 without loss of generality.In order to solve the Schrödinger equationH±ψ±n = E±n ψ±n (2)approximately, Marques et al. [1] chose the nonorthogonalbasis setφj (x) = x j−1e−|x|3/3, j = 1, 2, . . . (3)Table 3. Eigenvalues for the partner SUSY Hamiltonian operatorsfrom the Riccati-Padé method.Even (s = 0) Odd (s = 1)0 1.96950751375029482491.9695075137502948249 5.50717777714596996765.5071777771459699676 9.39426743787389146589.3942674378738914658 13.85837193654130014713.858371936541300147 18.64597563398844479918.645975633988444799 23.80718591798576691823.8071859179857669 29.232554550621496129.23255455062149608 34.946335718890610634.94633571889061and derived a single secular equation for both the evenand odd states. Since states of different parity do not mixthe dimension of their secular equation is twice the sizeof what would be actually necessary for a calculation ofthe same accuracy.In this paper we propose to work on the half line x ∈[0,∞) taking into account the appropriate boundary con-ditions at the origin: ψ(0) 6= 0, ψ ′(0) = 0 for the evenstates and ψ(0) = 0, ψ ′(0) 6= 0 for the odd ones. Theappropriate basis sets areφej = x je−x3/3, j = 0, 2, 3, . . . ,φoj = x je−x3/3, j = 1, 2, 3, . . . , (4)for the former and latter case, respectively.Tables 1 and 2 show Rayleigh-Ritz results for the firsteigenvalues of H− and H+, respectively, in terms of thesize N of the basis sets. We appreciate that the varia-tional results decrease with the number of basis functionsas expected. However, we are not aware of any proof thateither of the basis sets (3) or (4) is complete. We see thatpresent results are smaller, and therefore more accurate,than those of Marques et al. [1]. In addition, the resultsin their tables 1 and 2 exhibit unchanged entries whenthe size of the basis set increases from m to m+ 1 whichreflects the fact that the added basis function improvesthe even (odd) state but has no effect on the odd (even)one. As a result their matrices are twice the size of whatis necessary for a calculation of the same accuracy.The results of Tables 1 and 2, as well as those of Mar-ques et al. [1], suggest that E−n = E+n−1, n = 1, 2, . . ., asexpected for a SUSY pair of partner Hamiltonian opera-tors. However, those variational results are not sufficientlyaccurate and, as argued above, we are not aware of anyproof of convergence. In order to obtain more accurate re-sults we resort to the RPM [2–4]. To this end we consider471Accurate calculation of the eigenvalues of a new simple class of superpotentials in SUSY quantum mechanicsTable 4. Eigenvalues En for the quartic oscillator.N n = 0 n = 1 n = 22 1.077335422 3.804924324 8.1025312123 1.061889825 3.802939362 7.4864878664 1.060417725 3.800572274 7.4567380185 1.060362727 3.799746874 7.4565408806 1.060362727 3.799674368 7.4558264967 1.060362223 3.799673299 7.455703721RPM 1.0603620904841828996 3.7996730298013941688 7.4556979379867383922n = 3 n = 4 n = 52 11.867596773 11.77079816 16.91718138 21.922326894 11.67894913 16.29307103 21.798834885 11.64634787 16.29029658 21.285039336 11.64483615 16.26577116 21.241385537 11.64480669 16.26191320 21.24054440RPM 11.644745511378162021 16.261826018850225938 21.238372918235940024the modified logarithmic derivative of the wavefunction ψf (x) = sx − ψ ′(x)ψ(x) , (5)where s = 0 or s = 1 for even or odd states, respectively.It satisfies the Riccati equationf ′(x) = f (x)2 − 2sx f (x) + E − x4 ∓ 2x, x > 0 (6)and can be expanded in a Taylor series about the originf (x) = ∞∑j=0 fjxj , (7)where the coefficients fj can be easily obtained from theRiccati equation (6). We have f0 = −ψ ′(0)/ψ(0) = 0 forthe even states and choose f0 = 0 for the odd ones in orderto remove the unbalanced term 2f0/x that appears whenwe substitute the expansion (7) into the Riccati equation(6). Note that this application of the RPM to even po-tentials is different from the one in earlier papers [2–4]because in the present case we restrict the calculation tothe half line. The reason for such a modification is thatthe present even potentials cannot be expanded in an x2-power series.With the coefficients of the expansion (7) we construct theHankel determinants HdD = ∣∣fj+j+d−1∣∣Di,j=1 that are polyno-mial functions of the energy and obtain the approximateeigenvalues from the roots of HdD(E) = 0 [2–4]. More pre-cisely, we expect that sequences of roots E [D,d]n convergetowards the eigenvalues En, n = 0, 1, . . ., of the corre-sponding Hamiltonian operator as D increases. Since theSUSY partner potentials on the half line V±(x) = x4 ± 2xsatisfy V+(−x) = V−(x) then the RPM yields the eigen-values of both partner Hamiltonians simultaneously [5].More precisely, the sequences of roots for the even andodd wavefunctions will converge towards E−0 < E+0 <E−2 < E+2 < . . . and E−1 < E+1 < E−3 < E+3 < . . ., respec-tively.Table 3 shows the lowest eigenvalues for the SUSY part-ner Hamiltonians estimated from Hankel sequences withD ≤ 30, d = 0 and d = 1. The RPM yields the ground-state eigenvalue of H− exactly (E−0 = 0) and there is nodoubt from present accurate eigenvalues that E−n = E+n−1.Marques et al. [1] estimated the ground-state eigenvaluesfor the Schrödinger equation with the potentials V (x) = x4and V+(x) from the exact ground-state solution ψ−0 , E−0 forH−. They resorted to logarithmic perturbation theory buttheir results were not accurate. Here, we point out thatthey could have obtained more accurate results by meansof the variational method discussed above. In Table 4 weshow the results coming from the more efficient Rayleigh-Ritz variational method proposed in this paper and wealso add the accurate eigenvalues produced by the RPMfor comparison.3. ConclusionsIt is well known that the Rayleigh-Ritz variational methodis more efficient if one treats every symmetry species sepa-rately. In the present case we have carried out the separa-tion into even and odd functions on the half line by means472Francisco M. Fernándezof appropriate boundary conditions at origin. Thus, thesize of present secular equations, and consequently thecomputational effort for a given accuracy, is one half thatof Marques et al. [1].Working on the half line and taking into account theboundary conditions at origin also enables one to obtainhighly accurate eigenvalues by means of the RPM. In thepresent case one has to be cautious because the sameHankel determinant exhibits roots close to the eigenval-ues of both partner Hamiltonian operators. However, itis not difficult to identify the sequence of roots that con-verges to a chosen eigenvalue. The present RPM resultsconfirm that the two Hamiltonian operators proposed byMarques et al. [1] are already SUSY partners.The RPM was applied to sequences of SUSY partner po-tentials in the past [6]. However, the main interest in thatpaper was to obtain excited-state eigenvalues with Han-kel determinants of relatively small dimension. Besides,those potentials can be expanded in x2-power series andcan therefore be treated in the usual way.References[1] F. Marques, O. Negrini, A. J. da Silva, J. Phys. A 45,115307 (2012)[2] F. M. Fernández, Q. Ma, R. H. Tipping, Phys. Rev. A39, 1605 (1989)[3] F. M. Fernández, Q. Ma, R. H. Tipping, Phys. Rev. A40, 6149 (1989)[4] F. M. Fernández, arXiv:0807.0655v2 [math-ph][5] F. M. Fernández, J. Phys. A 29, 3167 (1996)[6] F. M. Fernández, Q. Ma, D. J. DeSmet, R. H. Tipping,Can. J. Phys. 67, 931 (1989)473

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Cent. Eur. J. Phys. • 11(4) • 2013 • 470-473DOI: 10.2478/s11534-013-0198-0Central European Journal of PhysicsAccurate calculation of the eigenvalues of a newsimple class of superpotentials in SUSY quantummechanicsResearch ArticleFrancisco M. Fernández∗INIFTA (UNLP, CCT La Plata-CONICET), División Química Teórica,Blvd. 113 S/N, Sucursal 4, Casilla de Correo 16,1900 La Plata, ArgentinaReceived 04 July 2012; accepted 17 February 2013Abstract: We obtain accurate eigenvalues for two recently derived SUSY partner Hamiltonians. We improve theRayleigh-Ritz variational method proposed by the authors and show how to apply the Riccati-Padé methodto those particular partner potentials.PACS (2008): 03.65.GeKeywords: SUSY partner Hamiltonians • variational calculation • Riccati-Padé method • accurate eigenvalues© Versita sp. z o.o.1. IntroductionIn a recent paper Marques et al. [1] derived two new simplesupersymmetric partner potentials and showed that theground-state eigenfunction of one of the Hamiltonian op-erators can be obtained exactly. In order to calculate theremaining states they resorted to the Rayleigh-Ritz vari-ational method. However, the convergence rate of theirapproach appears to be insufficient to reveal the rela-tion between the eigenvalues of the partner Hamiltoniansclearly.The authors also estimated the ground-state energy of thequartic anharmonic oscillator by means of perturbation∗E-mail: fernande@quimica.unlp.edu.artheory starting from the known solution for one of thepartner Hamiltonian operators.The purpose of this paper is twofold. First we improve thevariational method proposed by Marques et al. [1] and,second, we show that the Riccati-Padé method (RPM) [2–4] provides highly accurate results for the eigenvalues ofthose partner Hamiltonian operators. In addition, we showthat exactly the same Rayleigh-Ritz variational methodapplied to the SUSY partner Hamiltonians is also suit-able for the calculation of the eigenvalues of the quarticanharmonic oscillator.In Section 2 we apply the Rayleigh-Ritz variationalmethod and the RPM to the pair of SUSY Hamiltoni-ans proposed by Marques et al. [1] and also to the quarticoscillator. In Section 3 we comment on those approachesand draw conclusions.470Francisco M. Fernández2. SUSY partnersTable 1. Eigenvalues E−n from the Rayleigh-Ritz method.N n = 0 n = 1 n = 2 n = 3 n = 42 0 1.970246841 5.765408776 9.4885425543 0 1.970246137 5.511879703 9.488418664 14.130078374 0 1.969640134 5.507908922 9.406978612 13.892661955 0 1.969515816 5.507440820 9.394523525 13.874350606 0 1.969507628 5.507202146 9.394324659 13.859572597 0 1.969507538 5.507178381 9.394287127 13.85838650n = 5 n = 6 n = 7 n = 823 19.489629264 19.02107688 24.09989393 33.111071725 18.66009505 24.07406802 29.81927599 35.912983876 18.64895855 23.84777162 29.26028194 35.656564487 18.64704858 23.80790714 29.25748454 34.98577701Table 2. Eigenvalues E+n from the Rayleigh-Ritz method.N n = 0 n = 1 n = 2 n = 32 1.970246841 5.529040685 9.488542554 14.357212103 1.970246137 5.514418950 9.488418664 14.001035344 1.969640134 5.510107538 9.406978612 13.925448755 1.969515816 5.507493533 9.394523525 13.863892186 1.969507628 5.507185747 9.394324659 13.858512527 1.969507538 5.507178915 9.394287127 13.85851126n = 4 n = 5 n = 6 n = 723 19.48962926 24.528380464 19.02107688 24.52809061 33.11107172 37.633352295 18.66009505 23.91353748 29.81927599 36.331674416 18.64895855 23.81074205 29.26028194 35.025772447 18.64704858 23.81074194 29.25748454 34.98424967The partner Hamiltonian operators are given by [1]:H± = − d2dx2 + x4 ± 2|x|, (1)where we have chosen g = 1 without loss of generality.In order to solve the Schrödinger equationH±ψ±n = E±n ψ±n (2)approximately, Marques et al. [1] chose the nonorthogonalbasis setφj (x) = x j−1e−|x|3/3, j = 1, 2, . . . (3)Table 3. Eigenvalues for the partner SUSY Hamiltonian operatorsfrom the Riccati-Padé method.Even (s = 0) Odd (s = 1)0 1.96950751375029482491.9695075137502948249 5.50717777714596996765.5071777771459699676 9.39426743787389146589.3942674378738914658 13.85837193654130014713.858371936541300147 18.64597563398844479918.645975633988444799 23.80718591798576691823.8071859179857669 29.232554550621496129.23255455062149608 34.946335718890610634.94633571889061and derived a single secular equation for both the evenand odd states. Since states of different parity do not mixthe dimension of their secular equation is twice the sizeof what would be actually necessary for a calculation ofthe same accuracy.In this paper we propose to work on the half line x ∈[0,∞) taking into account the appropriate boundary con-ditions at the origin: ψ(0) 6= 0, ψ ′(0) = 0 for the evenstates and ψ(0) = 0, ψ ′(0) 6= 0 for the odd ones. Theappropriate basis sets areφej = x je−x3/3, j = 0, 2, 3, . . . ,φoj = x je−x3/3, j = 1, 2, 3, . . . , (4)for the former and latter case, respectively.Tables 1 and 2 show Rayleigh-Ritz results for the firsteigenvalues of H− and H+, respectively, in terms of thesize N of the basis sets. We appreciate that the varia-tional results decrease with the number of basis functionsas expected. However, we are not aware of any proof thateither of the basis sets (3) or (4) is complete. We see thatpresent results are smaller, and therefore more accurate,than those of Marques et al. [1]. In addition, the resultsin their tables 1 and 2 exhibit unchanged entries whenthe size of the basis set increases from m to m+ 1 whichreflects the fact that the added basis function improvesthe even (odd) state but has no effect on the odd (even)one. As a result their matrices are twice the size of whatis necessary for a calculation of the same accuracy.The results of Tables 1 and 2, as well as those of Mar-ques et al. [1], suggest that E−n = E+n−1, n = 1, 2, . . ., asexpected for a SUSY pair of partner Hamiltonian opera-tors. However, those variational results are not sufficientlyaccurate and, as argued above, we are not aware of anyproof of convergence. In order to obtain more accurate re-sults we resort to the RPM [2–4]. To this end we consider471Accurate calculation of the eigenvalues of a new simple class of superpotentials in SUSY quantum mechanicsTable 4. Eigenvalues En for the quartic oscillator.N n = 0 n = 1 n = 22 1.077335422 3.804924324 8.1025312123 1.061889825 3.802939362 7.4864878664 1.060417725 3.800572274 7.4567380185 1.060362727 3.799746874 7.4565408806 1.060362727 3.799674368 7.4558264967 1.060362223 3.799673299 7.455703721RPM 1.0603620904841828996 3.7996730298013941688 7.4556979379867383922n = 3 n = 4 n = 52 11.867596773 11.77079816 16.91718138 21.922326894 11.67894913 16.29307103 21.798834885 11.64634787 16.29029658 21.285039336 11.64483615 16.26577116 21.241385537 11.64480669 16.26191320 21.24054440RPM 11.644745511378162021 16.261826018850225938 21.238372918235940024the modified logarithmic derivative of the wavefunction ψf (x) = sx − ψ ′(x)ψ(x) , (5)where s = 0 or s = 1 for even or odd states, respectively.It satisfies the Riccati equationf ′(x) = f (x)2 − 2sx f (x) + E − x4 ∓ 2x, x > 0 (6)and can be expanded in a Taylor series about the originf (x) = ∞∑j=0 fjx j , (7)where the coefficients fj can be easily obtained from theRiccati equation (6). We have f0 = −ψ ′(0)/ψ(0) = 0 forthe even states and choose f0 = 0 for the odd ones in orderto remove the unbalanced term 2f0/x that appears whenwe substitute the expansion (7) into the Riccati equation(6). Note that this application of the RPM to even po-tentials is different from the one in earlier papers [2–4]because in the present case we restrict the calculation tothe half line. The reason for such a modification is thatthe present even potentials cannot be expanded in an x2-power series.With the coefficients of the expansion (7) we construct theHankel determinants HdD = ∣∣fj+j+d−1∣∣Di,j=1 that are polyno-mial functions of the energy and obtain the approximateeigenvalues from the roots of HdD(E) = 0 [2–4]. More pre-cisely, we expect that sequences of roots E [D,d]n convergetowards the eigenvalues En, n = 0, 1, . . ., of the corre-sponding Hamiltonian operator as D increases. Since theSUSY partner potentials on the half line V±(x) = x4 ± 2xsatisfy V+(−x) = V−(x) then the RPM yields the eigen-values of both partner Hamiltonians simultaneously [5].More precisely, the sequences of roots for the even andodd wavefunctions will converge towards E−0 < E+0 <E−2 < E+2 < . . . and E−1 < E+1 < E−3 < E+3 < . . ., respec-tively.Table 3 shows the lowest eigenvalues for the SUSY part-ner Hamiltonians estimated from Hankel sequences withD ≤ 30, d = 0 and d = 1. The RPM yields the ground-state eigenvalue of H− exactly (E−0 = 0) and there is nodoubt from present accurate eigenvalues that E−n = E+n−1.Marques et al. [1] estimated the ground-state eigenvaluesfor the Schrödinger equation with the potentials V (x) = x4and V+(x) from the exact ground-state solution ψ−0 , E−0 forH−. They resorted to logarithmic perturbation theory buttheir results were not accurate. Here, we point out thatthey could have obtained more accurate results by meansof the variational method discussed above. In Table 4 weshow the results coming from the more efficient Rayleigh-Ritz variational method proposed in this paper and wealso add the accurate eigenvalues produced by the RPMfor comparison.3. ConclusionsIt is well known that the Rayleigh-Ritz variational methodis more efficient if one treats every symmetry species sepa-rately. In the present case we have carried out the separa-tion into even and odd functions on the half line by means472Francisco M. Fernándezof appropriate boundary conditions at origin. Thus, thesize of present secular equations, and consequently thecomputational effort for a given accuracy, is one half thatof Marques et al. [1].Working on the half line and taking into account theboundary conditions at origin also enables one to obtainhighly accurate eigenvalues by means of the RPM. In thepresent case one has to be cautious because the sameHankel determinant exhibits roots close to the eigenval-ues of both partner Hamiltonian operators. However, itis not difficult to identify the sequence of roots that con-verges to a chosen eigenvalue. The present RPM resultsconfirm that the two Hamiltonian operators proposed byMarques et al. [1] are already SUSY partners.The RPM was applied to sequences of SUSY partner po-tentials in the past [6]. However, the main interest in thatpaper was to obtain excited-state eigenvalues with Han-kel determinants of relatively small dimension. Besides,those potentials can be expanded in x2-power series andcan therefore be treated in the usual way.References[1] F. Marques, O. Negrini, A. J. da Silva, J. Phys. A 45,115307 (2012)[2] F. M. Fernández, Q. Ma, R. H. Tipping, Phys. Rev. A39, 1605 (1989)[3] F. M. Fernández, Q. Ma, R. H. Tipping, Phys. Rev. A40, 6149 (1989)[4] F. M. Fernández, arXiv:0807.0655v2 [math-ph][5] F. M. Fernández, J. Phys. A 29, 3167 (1996)[6] F. M. Fernández, Q. Ma, D. J. DeSmet, R. H. Tipping,Can. J. Phys. 67, 931 (1989)473

Accurate calculation of the eigenvalues of a new simple class of superpotentials in SUSY quantum mechanics

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