A Convergent Renormalized Strong Coupling Perturbation Expansion for the Ground State Energy of the Quartic, Sextic, and Octic Anharmonic Oscillator

Weniger, Ernst Joachim

doi:10.1006/aphy.1996.0023

Cited by 106 publications

(99 citation statements)

References 47 publications

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“…3.4 and 3.5 is entirely based on the concept of distributional Borel summation and thus far from exhaustive. One aspect left out in this review concerns renormalized strong coupling expansions [306,307] and related methods (see also [102,108,111,112,[307][308][309][310]) which make possible the determination of very accurate numerical energy eigenvalues using only few perturbative coefficients, and which cover both small and large couplings. Expansions of this kind, however, fail for double-well like potentials and also cannot reproduce the width of resonances in the case of an odd perturbation.…”

Section: Discussionmentioning

confidence: 99%

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

et al. 2007

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This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is given by the singularities of the Borel transform, which introduce ambiguities from a mathematical point of view and lead to different possible physical interpretations. The two most important cases are: (i) the residues at the singularities correspond to the decay width of a resonance, and (ii) the presence of the singularities indicates the existence of nonperturbative contributions which cannot be accounted for on the basis of a Borel resummation and require generalizations toward resurgent expansions. Both of these cases are illustrated by examples.

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Section: Discussionmentioning

confidence: 99%

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

et al. 2007

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“…Here, one might argue that one should also investigate the summation of the perturbation expansion for the ground-state energy shift ∆E 0 (λ 2 ) with the help of the sequence transformations that were described in Section 7 and 8 of [32] and which produced very good results in the case of the anharmonic oscillators [30,33,34]. However, the convergence theory of these sequence transformations, which in the case of power series also produce rational approximants, is still very much in its infancy and no theoretical results concerning the transformation of Stieltjes series are known so far.…”

Section: Introductionmentioning

confidence: 99%

Numerical evidence that the perturbation expansion for a non-Hermitian PT-symmetric Hamiltonian is Stieltjes

Bender

Weniger

2001

Journal of Mathematical Physics

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102

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Recently, several studies of non-Hermitian Hamiltonians having PT symmetry have been conducted. Most striking about these complex Hamiltonians is how closely their properties resemble those of conventional Hermitian Hamiltonians. This paper presents further evidence of the similarity of these Hamiltonians to Hermitian Hamiltonians by examining the summation of the divergent weak-coupling perturbation series for the ground-state energy of the PT -symmetric Hamiltonian H = p 2 + 1 4 x 2 + iλx 3 recently studied by Bender and Dunne. For this purpose the first 193 (nonzero) coefficients of the Rayleigh-Schrödinger perturbation series in powers of λ 2 for the ground-state energy were calculated. Padé-summation and Padé-prediction techniques recently described by Weniger are applied to this perturbation series. The qualitative features of the results obtained in this way are indistinguishable from those obtained in the case of the perturbation series for the quartic anharmonic oscillator, which is known to be a Stieltjes series.

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“…It was shown in several articles that the transformation (A24) as well as its variant (A25) can be very effective [12][13][14][15][16][17]21,[23][24][25][26][27][29][30][31][62][63][64][65], in particular if strongly divergent alternating series are to be summed. In the case of the transformations (A22) and (A25), the approximation to the limit with the highest transformation order is given by…”

Section: Appendix A: Sequence Transformationsmentioning

confidence: 99%

“…Accordingly, there is an extensive literature on the summation of the divergent perturbation expansions of the anharmonic oscillators (see for example [14,[27][28][29][30][66][67][68][69][70][71][72] and references therein). In addition to the divergent weak coupling expansion (B3), there is also a strong coupling expansion [69] …”

Section: Appendix B: the Infinite Coupling Limit Of The Sextic Anharmmentioning

confidence: 99%

“…Moreover, numerous applications of sequence transformations have been reported in the literature. For example, the present author has applied sequence transformations successfully in such diverse fields as the evaluation of special functions [12][13][14][15][16][17], the evaluation of molecular multicenter integrals of exponentially decaying functions [18][19][20][21][22], the summation of strongly divergent quantum mechanical perturbation expansions [13,[23][24][25][26][27][28][29][30][31], and the extrapolation of crystal orbital and cluster calculations for oligomers to their infinite chain limits of stereoregular quasi-onedimensional organic polymers [32,33].…”

Section: Introductionmentioning

confidence: 99%

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Irregular input data in convergence acceleration and summation processes: General considerations and some special Gaussian hypergeometric series as model problems

Weniger

2001

Computer Physics Communications

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Sequence transformations accomplish an acceleration of convergence or a summation in the case of divergence by detecting and utilizing regularities of the elements of the sequence to be transformed. For sufficiently large indices, certain asymptotic regularities normally do exist, but the leading elements of a sequence may behave quite irregularly. The Gaussian hypergeometric series 2F1(a, b; c; z) is well suited to illuminate problems of that kind. Sequence transformations perform quite well for most parameters and arguments. If, however, the third parameter c of a nonterminating hypergeometric series 2F1 is a negative real number, the terms initially grow in magnitude like the terms of a mildly divergent series. The use of the leading terms of such a series as input data leads to unreliable and even completely nonsensical results. In contrast, sequence transformations produce good results if the leading irregular terms are excluded from the transformation process. Similar problems occur also in perturbation expansions. For example, summation results for the infinite coupling limit k3 of the sextic anharmonic oscillator can be improved considerably by excluding the leading terms from the transformation process. Finally, numerous new recurrence formulas for the 2F1(a, b; c; z) are derived.

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A Convergent Renormalized Strong Coupling Perturbation Expansion for the Ground State Energy of the Quartic, Sextic, and Octic Anharmonic Oscillator

Cited by 106 publications

References 47 publications

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

Numerical evidence that the perturbation expansion for a non-Hermitian PT-symmetric Hamiltonian is Stieltjes

Irregular input data in convergence acceleration and summation processes: General considerations and some special Gaussian hypergeometric series as model problems

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