On the maximum entropy method of handling divergent perturbation series

Volume 16 1, number 3

CHEMICAL PHYSICS LETTERS

15 September 1989

ON THE MAXIMUM ENTROPY METHOD OF HANDLING DIVERGENT PERTURBATION SERIES

Kamal BHATTACHARYYA Theory Group, Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India ’ and Department of Chemistry, Burdwan University,Burdwan 713 104, India Received 10 March 1989; in final form 24 May 1989

The recently introduced maximum entropy method of estimating divergent power-series sums has been employed in a more flexible form to extract reliable estimates of bounds to observable function values in rather hopeless cases, with very meage information about the series coefficients. The octic anharmonic oscillator and the hydrogenic Zeeman systems are studied as demonstrative examples. Results are also compared with a few other schemes.

1. Introduction Handling of divergent series [ 1 ] has been a longstanding problem in several branches of theoretical studies [ 2 1, in particular quantum mechanical perturbation theory [ 31. In these situations, it is of considerable interest to explore how far the various methods in vogue are effective in providing useful information from only the first few terms of a series, keeping in mind that large-order calculations for problems of practical concern often pose formidable technical difficulties. With low-order information, however, exact answers are impossible. So, naturally, one is tempted to seek for tight upper and lower bounds to the observable function values for which the series expansion form is derived, as otherwise computed data have to succeed some indirect scheme of assessment at the final step. Additionally, these bounds also offer us a good insight into the nature of variation, highly non-linear in character though owing to divergence, of the observable function with the coupling parameter, which, more often being physical in origin, is controllable and hence worth studying. In this respect, a special class of problems, leading to the so-called Stieltjes series, has become particularly appealing. The [n/n] and [(n- 1)/n] I Address for communication.

Padt approximants (PA) here provide a bracketing of the observable function [ 41. Remembering the intricacies of the conventional methods [S] of obtaining such bounds through other techniques, especially for observables other than the energy, this particular strategy of obtaining bounds from purely power-series expansions by proper manipulations seems to be extremely useful; it is quite convenient also in view of the calculational simplicity. For alternating series, for example, we have found [6] in this context that a computationally more efficient scheme than the PA is to adopt the generalised Euler transformation (GET) technique. A parametrised version of this transformation, called the parametrised Euler transformation (PET), provides through its [0,2] and [ 1,3] varieties of a two-parameter transform, in addition, even rough estimates of the presumed powerlaw dependence of the observable function on the coupling parameter [ 61. Very recently, a remarkably powerful method has emerged [7] from consideration of the maximum entropy principle [ 8,9] (MEP) whereby one obtains good lower bounds to functions defined formally by Stieltjes series. The basic idea involved in this scheme is to view the perturbation coefficients as the moments of a unknown probability distribution, an approximation to which is subsequently evaluated by employing the MEP. With increasing input infor-

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mation about these coefficients, such an approximation gradually improves, as is expected, leading ultimately to increasingly tighter bounds [ 7 1. Here, our endeavour in this context is threefold: firstly, to render the scheme more flexible to include nonStieltjes series as well; secondly, to investigate whether the method could also furnish useful upper bounds and finally, to numerically demonstrate the superiority of this approach over both the PA and the PET in bracketing the observable function.

2. The method Let us first consider an alternating “formal” powerseries representation of an unknown function F(t) of the form

(1) the coefficients of which are known to some fixed order, say k. It is called a Stieltjes series if F(z) admits an alternative integral representation:.

(2) where Q(x) is a bounded, non-decreasing measure function, hopefully differentiable, and p(x) will then be a non-negative function. Viewing the latter as a probability distribution, the expansion coefficients s} in ( 1) may be identified as various moments of p(x) which are finite and real valued &jp(x)x’dx,

j=O, 1,2, ._.k,

(3)

0

with the lowest

one u-0) dictating the proper normalisation of the distribution function p(x). The point here is, if a straightforward summation of ( 1) turns out to be meaningless owing to divergence, and if it is a Stieltjes series, one can proceed through (2 ) to evaluate F(z) after having found a suitable strategy of constructing p(x). But then, a number of problems emerge immediately. Thus, one has to consider (i) whether ( 1) corresponds really to a Stieltjes series, (ii) whether there is a way of constructing p (x ) approximately from an incomplete knowledge

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of G} which could be improved systematically with increasing information about the coefficients and finally (iii) whether the constructed p(x) is unique. Deferring a discussion on (i ) for the present, first we like to remark that, although it is possible to construct various approximations to p(x) which would satisfy (3), the “least biased” estimate [ 91 is provided by the MEP. As regards (iii), again an inherent non-uniqueness in p(x) would develop if the Carleman condition [ 41 ceases to hold; yet, also then one can proceed with the MEP solution of p(x), as sketched in ref. [ 71, in view of its “most likely” character. Now, coming back to (i), we note that if the analytic structure of P(z) as a function of the complex variable z is known, one might assert [ 71 whether F(r) is a Stieltjes function For real z (O< z< 00) also, a similar conclusion follows (e.g. theorem 5.5.3 in ref. [4]). On the other hand, straightforward route of constructing the Hadamard determinants D (m, n ) from the known a} and verifyingwhetherD(O,n)>OandD(l,n)>Oforalln (theorem 5.3.1. in ref. [4] ) is appropriate only to disprove the Stieltjes nature [ 10 1. However, the necessary conditions for the simultaneous satisfaction of ( 1) and (2 ) may here be noted, F(z)aO,

o~zscn)

hH/h2~/fb)2,’

n=1,2 ,.a..

(4)

The first of these conditions follows from the positive semidefiniteness of the integrand in (2) while the second one is based on the familiar inequality for the mean squared deviation of a property like x”, given by (x2”> 2 (x”) 2 where “( )” refers to statistical averaging with a normalised (to unity) distribution, p(x)/S, in the present case. For the discussion to follow, (4) would turn out to be an important guide. The MEP solution forp(x) in (2) is given by [ 7,9]

(5) #} in ( 1) is known to order k. The unknown Lagrange multipliers Ai are determined by requiring (5 ) to satisfy ( 3 ) wherefrom k+ 1 auxiliary conditions would emerge. In general, however, this nonlinear system of equations is difficult to solve [ 9,11,12] ; for k> 1, an analytical solution is actually when

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impossible and, as k increases, numerical instability gradually grows up to soon become unmanageable. Thus, this MEP method which rests on the relation m F(z)=

p&c)

(1 +zX)-’ dX + O(zk+‘)

0

=MEP(k)+

U(Z’+‘)

(6)

and relies on a smooth passage towards the k+cc limit is particularly well suited when only the first few J are known. In ref. [ 71, the octic anharmonic oscillator case has been studied with k= 1, 3 and 5. They provide gradually improved lower bounds to the ground-state energy. But, one observes here also that the betterment with increasing information (or k) is quite poor. In other words, (6) becomes progressively insensitive to a change in k. This is another important aspect which has led us to explore ways of generalising ( 2). Our strategy is based on viewing the representation (2) as an expectation value problem for a given positive definite multiplicative operator. Then, increasing k corresponds to employing gradually better probability functions &(x), or, to state it otherwise, wavefunctions, for p(x). It is known, however, that approximation at the operator level is usually much more sensitive than at the wavefunction level. We thus anticipate that, instead of (6), the following more general scheme: F(Z)=

I 0

pj(X)Ak-j(Z,X)

=GMEPQ,k)+Lo(zk+*),

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CHEMICAL PHYSICS LETTERS

dX+

k should here be hopefully prominent because a deliberate change at the operator level is possible, keeping pi(X) fixed at some chosen j. To avoid any numerical instability that one faces in the course of implementing (6 ), we have here restricted ourselves to the first two simplest possibilities: j= 1 and k= 1, 2. For k= 2, Ai( z, x) has been chosen as ( 1+zx) -01, CYbeing an adjustable real parameter. Such a choice in (7) actually provides two additionally remarkable features. The first of these is concerned with the Stieltjes series; here, whereas the k= 1 case yields lower bounds, good upper bounds are obtained with k= 2, as the examples considered below will show. In accordance with (7), one chooses a! in such a way that the integral representation in (7) would agree with the power-series form ( 1) of I;(z) to O(zz). Thus, in this case, the agreement between the two equivalent forms for F(z) is one order better than the former one for k= 1, with cy= 1, i.e. the parent MEP scheme or GMEP( 1, 1). Let us note here also that these two GMEP( 1, k) are rather easy to evaluate, requiring just a good numerical integration procedure for an integral of the form N

I

(l+zx)-“exp(-/3x)d_x,

0

where N=exp( -A,) in (5), and the parameters LY, /I and N are related to the expansion coefficients of F(z) in (1) by GEMP(l, 1):

U(Zkf')

~y=l ,

B=hlfi , jai,

(7)

with pj(X) having the same form as (5) and Ai referring to a multiplicative, positive definite operator containing i embedded parameters (A0=s( 1-t ZX)- ’ ) , would perform far more efficiently. Indeed, introduction of such a flexibility is of considerable significance owing to the following observations: (i) The choice j= k leads us back to the parent prescription (6). (ii) The j= 1 case refers to the other extreme possibility where p(x) would assume a simple exponential form and here, notably, if the condition A (z, X) > 0 is relaxed to allow a power-series form for A, the more familiar Bore1 transform [ 131 of ( 1) would show up. (iii) Improvement with increasing

GMEP( L2):

a=f:l

N=

Ph,

U&h -f:)

B=orfo/fi,

,

N=BsO.

(8)

The second advantage with the parameter (Yis, even a simple non-Stieltjes series like F(z) = ( 1 +z) -* = 1-2z+3z2-..., which fails to satisfy the moment constraints in (4) and is hence not summable through (6), may well be handled now by choosing CY>2. In fact, here the form of the probability function (exponential) would actually force us to take some cyr 2, as otherwise, at (Y= 2, the distribution has to be very special, dispersion free. The strategy in such cases would hence be quite different from those of the Stieltjes ones. Thus, GMEP( 1, 2), with 261

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(Yr 2 would agree here with F(z) to U( z) . However, one may optimise the cyvalue, if possible, by ensuring minimisation of I W( z)/6a I since F(z) should ideally be independent of CY.Thus, this generalised MEP (GMEP) scheme given by (7) possesses a number of desirable qualifications.

3. Results and discussion To demonstrate the workability of the GMEP scheme and its superiority over the PA and the PET, here a few selected perturbation series are studied. All these series have one feature in common: they diverge very badly. In fact, our choice has been primarily guided by this particular property of such series because these strongly divergent situations really provide the proper testing ground for assessing the worth of any new strategy. Thus, attempts to extract reasonable estimates of function values from these series have also drawn considerable recent attention. We first consider the ground-state energy series for the octic anharmonic oscillator [ 71 given by the Hamiltonian H= - ~d2/dxz+ 4x2-t zx8, for which exact results are also available [ 141. Here E,(z) = f +zF(z), and GMEP (j, k), as the integral part at the rhs in (7) prescribes, is applied on F(z). Similar is the case with the PA which requires the same knowledge of the coefficients. The PET, however, is directly employed on &(z) . In table 1, all these results are displayed. Remembering that z= 1 really corresponds to a very strong perturbation and that one obtains the results from a knowledge of the first three perturbation terms only, we find the bracketing quite encouraging. Here, it is known that F(z) is a Stieltjes series but the Carleman condition is not satisfied and hence the PA results do not con-

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verge even if one uses information of many more perturbationterms. Thus, at z= 1, [9/9] and [9/10] PA bracket the exact value very loosely; the results are 4.3316 and 0.5414 respectively [7]. The strength of the PET is now seen very clearly. At the same time, the spectacular performance of the GMEP method is also readily established. Lower bounds here provide closest approximations. We also note the significant change in passing from GMEP( 1, 1) to GMEP ( 1, 2) which may be compared with the changes in going from MEP ( 1) to MEP ( 5 ) through MEP (3 ), as given in ref. [ 7 1. This observation justifies the basis of prescription (7). Tables 2 and 3 deal with energy shifts for the 1s and 2s states, respectively, of the H atom in a magnetic field. This perturbed system has received considerable recent attention [ 15- 18 ] from a calculational standpoint, probably more because of its added cosmological relevance [ 191. The Hamiltonian for the problem is H= -id-l/r+~z(x2+y2). Here also, z= 1 refers to a very large perturbation. Indeed, magnetic fields of actual experimental concern, limited to lo6 G or so, fall well within z=O.l . To implement the GMEP or the PA, one here defines P(z) byAE(z)=E(z)-e=zF(z),withestandingforthe unperturbed energy, and finds that it satisfies (4). The same knowledge about the series coefficients [ 161 (to fhird order) is also utilised to approximate E(z) by the PET. The results, as before, point to increasing efficiency along the order PA, PET, GMEP. For the ground state, however, one observes no startling improvement. This exhibits another characteristic feature of such a comparative study: stronger divergence would always lead to a larger strategic difference. Thus, milder divergence for the 1s state accounts for the success of several large-order calculational schemes like the GET [ 17 1, the order-de-

Table 1 Ground-state energy values for the octic anharmonic oscillator z

IO/ 1IPA

[O,Z]PET

GMEP(l,l)

Exact

GMEP(l,Z)

[1,3]PET

[l/I]PA

0.000 I

0.50064 0.50497 0.51557 0.51980 0.52029 0.52035

0.50064 0.50507 0.52065 0.5438 1 0.56147 0.56929

0.50064 0.50519 0.52449 0.56157 0.59261 0.60645

0.50064 0.50543 0.53210 0.6205 I 0.7455 1 0.82069

0.50064 0.50551 0.53764 0.71687 1.20792 1.67348

0.50064 0.50561 0.54391 0.83349 1.87243 3.02377

0.50064 0.50571 0.55233 1.01530 3.07272 5.64449

0.001 0.01 0.1 0.5

1

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Table 2 Energy shifts for the ground state of spinless hydrogenic Zeeman system z

[O/lIPA

[O,Z]PET

GMEP(1,l)

Exact

GMEP(l,Z)

[ 1,3]PET

11/l

0.0001 0.01 0.04 0.36

0.000025 0.002473 0.009577 0.064401 0.118812

o.oooo25 0.002473 0.009580 0.065448 0.126470

0.000025 0.002473 0.009593 0.068381 0.144293

0.000025 0.002474 0.0096 18 0.072538 0.168831

0.000025 0.002474 0.0096 19 0.072750 0.170645

0.000025 0.002474 0.009622 0.074374 0.183195

0.000025 0.002474 0.009624 0.076131 0.198763

I

IPA

Table 3 Energy shifts for the 2s state of spinless hydrogenic Zccman system 2

K’/llPA

[O,Z]PET

GMEP( 1, 1)

Exact

GMEP( 1,2)

[ 1,3]PET

Il/llPA

0.0001 0.01 0.04 0.36 1

0.000348 0.02405 1 0.049629 0.072461 0.075230

0.000348 0.024395 0.053537 0.094352 0.105293

0.000348 0.025836 0.067138 0.186279 0.256430

0.000348 0.0269 11 0.07601 0.27223 0.46456

0.000348 0.027064 0.079418 0.336583 0.613610

0.000348 0.027715 0.089153 0.535163 1.223435

0.000348 0.028396 0.101668 0.861789 2.380035

pendent mapping method of Le Guillou and ZinnJustin [ 18 ] and even the PA [ 15 1. Incidentally, convergence of the PA in this case at z= 1 [ 15 ] implies, among others, the Stieltjes nature of F(z). On the other hand, excited states diverge much more strongly and hence have attracted little attention [ 161. But, we find accordingly in table 3 a remarkable improvement in going either from the PA to the PET or from the PET to the GMEP for the 2s state. The 2p,, state has also been studied in ref. [ 16 1. Divergence in this case is somewhat less severe than that of the 2s state and, as expected, bracketing of AE(z) follows here a similar trend. So, instead of presenting detailed results, we just quote the data at z= 1, for example, where the exact value (0.6684) is bounded by three couples: 0.0756, 2.1749 (PA); 0.1043, 1.2417 (PET); 0.2471, 0.7023 (GMEP). Anyway, it is not merely true that only good bounds are provided by the GMEP, as these examples demonstrate; the closer estimates, given by the upper bounds for this system, may even be comparable with other large-order results also. For example, the functional method [ 201 (FM) of ref. [ 161, though quite successful for the 1s or 2p,, states, furnishes relatively poor quality data for the 2s state compared to.the GMEP( 1, 2). This is clearly seen from table 3 and the FM estimates [ 161: 0.02727 (z=O.Ol); 0.8125 (~~0.04); 0.3483 (z=O.36); 0.6260 (t= 1). Rememberingthat theFM

makes use of as much as 13 perturbation terms, whereas the GMEP( 1,2) involves only 3 such terms, the enormous strength of the present strategy may be appreciated without any reservation. The above observation also indicates that the GMEP scheme is inherently much more balanced than usual series acceleration methods like the FM.

4. Conclusion The primary aim of our study has been to render the MEP scheme of handling divergent power-series expansions more flexible in order that a few additionally desirable features could be incorporated to gain more advantage. We hope to have achieved the end through the GMEP. Thus, this new scheme provides upper bounds for Stieltjes-like series, permits handling of at least a class of non-Stieltjes series as well and changes significantly with increasing input information. Calculational demonstrations also make it quite evident that the bounds presented by this scheme are far superior to any other familiar procedure. Moreover, in some cases, the resulting closer bounds are found to be even better than a few other estimates involving a knowledge of many more perturbation terms. Thus, an exemplary performance of this GMEP scheme has been established here, re263

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membering the little computational labour and the meagre input information involved. Finally, we hope that further work along this line will also turn out to be rewarding and prove the value of the method, especially in so far as the bracketing property, which the present numerical experience have led us to conjecture, is concerned.

References [ k] K. Knapp, Theory and application ofinfinite series (Blackie, London, 1928); G.H. Hardy, Divergent series (Oxford Univ. Press, Oxford, 1956). [2] G.A. Baker Jr. and J.L. Gammel, eds., The Padt approximant in theoretical physics (Academic Press, New York, 1970); C. Domb and MS. Green, eds., Phase transitions and critical phenomena, Vo13 (Academic Press, New York, 1974); C.J. Pearce, Advan. Phys. 27 (1978) 89. [ 31 J. Zinn-Justin, Phys. Rept. 70 ( 198 1) 109; Intern. J. Quantum Chem. 2 1 (1982). [ 41 GA. Baker Jr. and P. Graves-Morris, Padt approximants, Part I (Addison-Wesley, Reading, 1981) . [ 51F. Weinhold, Advan. Quantum Chem. 6 (1972) 299; M.A. Abdel-Raouf, Phys. Rept. 84 (1982) 163; R. Balian andM. Veneroni, Ann. Phys. NY 187 (1988) 29. [6] K. .Bhattacharyya, Intern. J. Quantum Chem. 22 (1982) 307; Proc. Indian Acad. Sci. Chem. Sci. 99 (1987) 9.

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[7] C.M. Bender, L.R. Mead and N. Papanicolaou, J. Math. Phys. 28 (1987) 1016. [S] E.T. Jaynes, Phys. Rev. 106 (1957) 620; 108 (1957) 171; R.D. Levine and M. Tribus, eds., The maximum entropy formalism (MIT Press, Cambridge, 1979 ), [ 91 L.R. Mead and N. Papanicolaou, J. Math. Phys. 25 ( 1984) 2404. [ lo] E.R. Vrscay, Phys. Rev. A 33 (1986) 1433. [ 111 N. Agmon, Y. Alhassid and R.D. Levine, J. Comput. Phys. 30 (1979) 250. [ 121 W. Jaworsky and R.S. Ingarden, Acta. Phys. Polon. A 59 (1981) 635. [ 131 B. Simon, Intern. J. Quantum Chem. 21 (1982) 3. [ 141 ET. Hioe, D. MacMiien and E.W. Montroll, J. Math. Phys. 17 (1976) 1320. [IS] J. Cifek and E.R. Vrscay, Intern. J. Quantum. Chem. 21 (1982) 27. [ 161 G.A. Arteca, EM. Femandez, A.M. Meson and E.A. Castro, Physica A 128 (1984) 253. [ 171J.N. Silverman, Phys. Rev. A 28 ( 1983) 498. [ 181J.C. Le Guillou and J. Zinn-Justin, Ann. Phys. NY 147 (1983) 57; G. Rosen, Phys. Rev. A 34 ( 1986) 1556; C.R. Handy, D. Bessis, G. Sigismondi and T.D. Morley, Phys. Rev. Letters 60 (1988) 253. [ 191 R.H. Garstang, Rept. Progt. Phys. 40 ( 1977) 105. [20] G.A. Arteea, F.M. Femandez and E.A. Castro, Chem. Phys. Letters 102 (1983) 344; Physica A 128 (1984) 589; J. Math. Phys. 25 ( 1984) 3492.