Acta Polytechnica doi:10.14311/AP.2017.57.0391 Acta Polytechnica 57(6):391–398, 2017 © Czech Technical University in Prague, 2017 available online at http://ojs.cvut.cz/ojs/index.php/ap HIGHLY ACCURATE CALCULATION OF THE REAL AND COMPLEX EIGENVALUES OF ONE-DIMENSIONAL ANHARMONIC OSCILLATORS Francisco Marcelo Fernández∗, Javier Garcia INIFTA (UNLP, CCT La Plata-CONICET), Blvd. 113 S/N, Sucursal 4, Casilla de Correo 16, 1900 La Plata, Argentina ∗ corresponding author: fernande@quimica.unlp.edu.ar Abstract. We draw attention on the fact that the Riccati-Padé method developed some time ago enables the accurate calculation of bound-state eigenvalues as well as of resonances embedded either in the continuum or in the discrete spectrum. We apply the approach to several one-dimensional models that exhibit different kind of spectra. In particular we test a WKB formula for the imaginary part of the resonance in the discrete spectrum of a three-well potential. Keywords: anharmonic oscillators; bound states; resonances; Riccati-Padé method; WKB asymptotic expression. 1. Introduction In a recent paper Gaudreau et al[1] proposed a method for the calculation of the eigenvalues of the Schrödinger equation for one-dimensional anharmonic oscillators. In their analysis of some of the many approaches proposed earlier with that purpose they resorted to expressions of the form: “However, the existing numerical methods are mostly case specific and lack uniformity when faced with a general problem.” “As can be seen by the numerous approaches which have been developed to solve this problem, there is a beautiful diversity yet lack of uniformity in its resolution. While several of these methods yield excellent results for specific cases, it would be favorable to have one general method that could handle any anharmonic potential while being capable of computing efficiently approximations of eigenvalues to a high pre-determined accuracy.” “Various methods have been used to calculate the energy eigenvalues of quantum anharmonic oscillators given a specific set of parameters. While several of these methods yield excellent results for specific cases, there is a beautiful diversity yet lack of uniformity in the resolution of this problem.” The authors put forward an approach that they termed double exponential Sinc collocation method (DESCM) and reported results of remarkable accuracy for a wide variety of problems. In fact they stated that “In the present work, we use this method to compute energy eigenvalues of anharmonic oscillators to unprecedented accuracy” which may perhaps be true for some of the models chosen but not for other similar examples. For example, in an unpublished article Trott[2] obtained the ground-state energy of the anharmonic oscillator with potential V (x) = x4 with more than 1000 accurate digits. His approach is based on the straightforward expansion of the wavefunction in a Taylor series about the origin. One of the methods mentioned by Gaudreau et al[1] is the Riccati-Padé method (RPM) based on a rational approximation to the logarithmic derivative of the wavefunction that satisfies a well known Riccati equation[3, 4]. In their brief analysis of the RPM the authors did not mention that this approach not only yields the bound-state eigenvalues but also the resonances embedded in the continuum[5]. What is more, the same RPM quantization condition, given by a Hankel determinant, produces the bound-state eigenvalues, the resonances embedded in the continuum as well as some kind of strange resonances located in the discrete spectrum of some multiple-well oscillators[6]. It is not clear from the content of[1] whether the DESCM is also suitable for the calculation of such complex eigenvalues. The accuracy of the calculated eigenvalues not only depends on the chosen method but also on the available computation facilities and on the art of programming. For this reason the comparison of the accuracy of the results reported in a number of papers spread in time should be carried out with care. The purpose of this paper is two-fold. First, we show that the RPM can in fact yield extremely accurate eigenvalues because it exhibits exponential convergence. To that end it is only necessary to program the quantization condition in an efficient way in a convenient platform. Second, we stress the fact that the RPM yields both real and complex eigenvalues with similar accuracy through the same quantization condition. More precisely: it is not necessary to modify the algorithm in order to obtain such apparently dissimilar types of eigenvalues that are associated to different boundary conditions of the eigensolution. In section 2 we outline the RPM for even-parity potentials. In section 3 we apply this approach to some of the examples discussed by Gaudreau et al[1] and obtain eigenvalues with remarkable accuracy. In this section we also calculate several resonances supported by anharmonic oscillators that were not taken into account by those authors. 391 http://dx.doi.org/10.14311/AP.2017.57.0391 http://ojs.cvut.cz/ojs/index.php/ap Francisco Marcelo Fernández, Javier Garcia Acta Polytechnica We consider examples of resonances embedded in the continuous as well as in the discrete spectrum. Finally, in section 4 we summarize the main results and draw conclusions. 2. The Riccati-Padé method The dimensionless Schrödinger equation for a one-dimensional model reads ψ′′(x) + [E −V (x)]ψ(x) = 0, (1) where E is the eigenvalue and ψ(x) is the eigenfunction that satisfies some given boundary conditions. For example, lim |x|→∞ ψ(x) = 0 determines the discrete spectrum and the resonances are associated to outgoing waves in each channel (for example, ψ(x) ∼ Aeikx). In this paper we restrict ourselves to anharmonic oscillators with even-parity potentials V (−x) = V (x) to facilitate the comparison with the results reported by Gaudreau et al[1] but it should be taken into account that the approach applies also to no non-symmetric potentials[7]. In order to apply the RPM we define the regularized logarithmic derivative of the eigenfunction f(x) = s x − ψ′(x) ψ(x) , (2) that satisfies the Riccati equation f′(x) + 2sf(x) x −f(x)2 + V (x) −E = 0, (3) where s = 0 or s = 1 for even or odd states, respectively. If V (x) is a polynomial function of x or it can be expanded in a Taylor series about x = 0 then one can also expand f(x) in a Taylor series about the origin f(x) = x ∞∑ j=0 fj (E)x2j. (4) On arguing as in earlier papers we conclude that we can obtain approximate eigenvalues to the Schrödinger equation from the roots of the Hankel determinant HdD (E) = ∣∣∣∣∣∣∣∣∣ fd+1 fd+2 · · · fd+D fd+2 fd+3 · · · fd+D+1 ... ... ... ... fd+D fd+D+1 · · · fd+D−1 ∣∣∣∣∣∣∣∣∣ = 0, (5) where D = 2, 3, . . . is the dimension of the determinant and d is the difference between the degrees of the polynomials in the numerator and denominator of the rational approximation to f(x)[3–6]. In those earlier papers we have shown that there are sequences of roots E[D,d], D = 2, 3, . . . of the determinant HdD (E) that converge towards the bound states and resonances of the quantum-mechanical problem. We have at our disposal a set of sequences for each value of d but it is commonly sufficient to choose d = 0. For this reason, in this paper we restrict ourselves to the sequences of roots E[D] = E[D,0] (unless stated otherwise). In this paper we are concerned with anharmonic-oscillator potentials of the form V (x) = K∑ j=1 vjx 2j. (6) The spectrum is discrete when vK > 0 and continuous when vK < 0. In the latter case there may be resonances embedded in the continuous spectrum which are complex eigenvalues. The real part of any such eigenvalue is the resonance position and the imaginary part is half its width Γ (|=E| = Γ/2). 3. Examples Four examples chosen by Gaudreau et al[1] are quasi-exactly solvable problems; that is to say, one can obtain exact solutions for some states: V1(x) = x2 − 4x4 + x6 E0 = −2 V2(x) = 4x2 − 6x4 + x6 E1 = −9 V3(x) = 105 64 x2 − 43 8 x4 + x6 −x8 + x10 E0 = 3 8 V4(x) = 169 64 x2 − 59 8 x4 + x6 −x8 + x10 E1 = 9 8 . (7) 392 vol. 57 no. 6/2017 One-dimensional anharmonic oscillators The RPM yields the exact result for all these particular cases because in all of them the logarithmic derivative f(x) is a rational function of the coordinate. The Hankel determinants of lowest dimension for each case are: H02 (E) = 1 4725 (E + 2)(E5 − 2E4 − 23E3 − 602E2 + 1030E − 1412), (8) H02 (E) = 1 4465125 (E + 9)(E5 − 9E4 − 187E3 − 8217E2 + 78336E − 348624), (9) H03 (E) = 1 3189612751764848640000 (8E − 3)(8589934592E11 + 3221225472E10 − 1887235473408E9 − 399347250364416E8 − 1634745666502656E7 + 10770225531715584E6 − 836065166572191744E5 − 905684630058491904E4 + 5197219286067104256E3 − 2944302537136698432E2 − 12283878786837315912E + 22452709866105906693), (10) H03 (E) = 1 431028319209742820966400000 (8E − 9)(8589934592E11 + 9663676416E10 − 5569096187904E9 − 2064531673055232E8 − 15362232560910336E7 + 158709729905344512E6 − 23752960275863896064E5 − 84068173973645402112E4 + 2318080070178601634304E3 − 6274577633554290840768E2 − 75410626140297229262472E + 655367638076442656931879), (11) respectively. It is clear that the second factor of each Hankel determinant yields the exact eigenvalue of the corresponding model in equation (7). As a nontrivial example we consider the quartic anharmonic oscillator V (x) = x2 + λx4. (12) Gaudreau et al [1] calculated the ground state for λ = 1 with remarkable accuracy. The RPM also enables great accuracy because of its exponential convergence. For example, with determinants of dimension D ≤ 623 we obtained E0 = 1.39235164153029185565750787660993418460006671122083408890634932387756743187564652 85909735634677917591211513753417388174455516240463837130438178697370013460935168 15484208574889656901800305541236648743218953435715417409382624057229519998568711 18140968922702273638169811112603107034293861341959645684859182914634898518858148 63025469392145221031177208948219643654580541741801366088701870825264349698158700 82340760759574319226851138960019685449394982096240756162094619633463447377455701 49211492623468905916373385630626814055709925106270580909505786666030935831448351 97352905560061049224302849821825415119194035000689109989896675454979833183805654 19975466162573031052729404581567529262538228672118076018319975294595611113245756 78445653018419567798509749315372254188588216960225999726980950846580656370213654 47651793869049904755455309191949465274340562585980971938979595684138772300267900 68177673277845708654477245631366268184519934644126051969150124972306172724393638 74511499751517142498813649966422950045954851519165072488133686158144218817306000 39773536840117104637678735672726392478420532548924901523470626991951440934018875 83071929546817823113125377471312004221881276679422460872268510606766179549130792 640798558850522732484547554994100518213983, which is considerably more accurate than the result reported by those authors. Such an accuracy is unnecessary but it clearly shows the stability and remarkable rate of convergence of the RPM. For some approaches the pure quartic oscillator V (x) = x4 (13) 393 Francisco Marcelo Fernández, Javier Garcia Acta Polytechnica may be more demanding than the oscillator treated previously. However, this is not the case for the RPM that yields E0 = 1.06036209048418289964704601669266354551520872852829779332162452416959435630443444 21126896299134671720351054624435858252558087980821029314701317683637238249357892 26246004708175446960141637488417282256290593575779088806178879026360154939569027 51961489200942934873584409442694897901213971464290951923352453382834703350575761 51120257039888523720240221842110308657373109139891545365841031116794058335486020 09227440069631126702388622971429699610592155832226671376935508673610000831830027 51792623357391390621361807764985969618149941279280927284070795610604240722946809 94913627572927387279136890279842472226217169444889547513704380684054391877877295 32342458274372543178323190603810687416044034374530146847272813918612940470431034 01351071607110353008929823272542766151898695056504716025275608952626219102568822 00964410287815640052705292932405076382650282591122477362538471854714402572285438 48529745045857097828402490669995704768445877091762029124375273254907211643344023 02947306923981908956853745359884460160202313291933059395869304916644281633946163 32428700242614612377430099522342042085977356901535654168502308941851348795734106 58547971946759646679661346762885864379526545195605682867159583388847434670120422 42071491874787103842957338913898524589402226347126961769965604409311709985471606 46641857421281143028818111495112214843140887121662059313076923418022298272468836 26045356507913236221596486925870033200274440968806404623978817839469837807048268 60217427219460350750696191658224983009606134572666392863592217643534013718920448 14846483730289412529638634402446954353934473733433447707230478215508820964235121 06900382833900237848230939194834 from determinants of dimension D ≤ 806. We think that present result is more accurate than the one reported by Trott[2], the discrepancy being in his last 9 figures. In the case of the quartic anharmonic oscillators (6) and (12) it was proved that E[D,0]0 and E [D,1] 0 are lower and upper bounds to the actual eigenvalue, respectively[3, 4]. In order to verify the accuracy of present calculation we verified that both bounds agreed to the last digit. As stated in the introduction, the RPM yields not only the bound states but also the resonances. For example, in the case of the anharmonic oscillator (12) with λ = −0.1 (inverted double well) we obtain