Complex energies from real perturbation series for the LoSurdo-Stark effect in hydrogen by Borel-Padé approximants

Franceschini, Valter; Grecchi, V.; Silverstone, Harris J.

doi:10.1103/physreva.32.1338

Cited by 82 publications

(51 citation statements)

References 31 publications

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“…[207][208][209]. These contours are just straight lines tilted in the complex plane so that the singularities of the Padé approximants are appropriately encircled when Jordan's Lemma is applied, to give the pole contributions that finally yield numerical approximations to the imaginary part of the Borel sum.…”

Section: Borel-padé Methodsmentioning

confidence: 99%

“…[207][208][209], it has been shown that integrations along the contours C j have been shown to provide the physically correct analytic continuation of resummed perturbation series for those cases where the evaluation of the standard Laplace-Borel integral is impossible due to multiple branch cuts or due to spurious singularities in view of the finite order of the Padé approximant (196). This is confirmed in the current review via the example of the cubic anharmonic oscillator treated in Sec.…”

Section: Borel-padé Methodsmentioning

confidence: 99%

“…Before starting the numerical analysis of the cubic potential, however, we may mention that the problem of "recovering" the complex resonance energies from a nonalternating perturbation series has been discussed for the Stark effect in [207,209] and for the quantum electrodynamics (QED) effective action in [208]. For the Stark effect, for example, the situation is as follows: the problem is spectrally unstable: as soon as the electric field is turned on, all eigenvalues of the hydrogen atom disappear in the continuous spectrum (a phenomenon first proven by Titchmarsh [233]).…”

Section: Theorem 13mentioning

confidence: 99%

“…(206), even for small coupling g. Therefore, we have to apply special resummation techniques in order to recover the partition function Z(g) from the divergent series (207). In our test calculations, we have applied four resummation techniques: (i) the δ transformation defined in Eq.…”

Section: Zero-dimensional Theorymentioning

confidence: 99%

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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: Borel-padé Methodsmentioning

confidence: 99%

Section: Borel-padé Methodsmentioning

confidence: 99%

Section: Theorem 13mentioning

confidence: 99%

Section: Zero-dimensional Theorymentioning

confidence: 99%

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

et al. 2007

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“…Solutions for expression (41) are the energies obtained using the WKB approximation for a half wedge potential with an infinitely high wall at x = −a. Therefore, we interpret the energies satisfying (41) as those of the backscattering states between the left well and the electric field, which is the Ramsauer effect illustrated in Figures 5 and 6.…”

Section: Analytic Properties Of Spectral Function: Stark Shifts and Lmentioning

confidence: 99%

Simple soluble molecular ionization model

Dunne

Gauthier

2004

Phys. Rev. A

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We present a simple exact analytical solution, using the Weyl-Titchmarsh-Kodaira spectral theorem, for the spectral function of the one-dimensional diatomic molecule model consisting of two attractive delta function wells in the presence of a static external electric field. For sufficiently deep and far apart wells, this molecule supports both an even and an odd state, and the introduction of a static electric field turns these bound states into quasi-bound states which are Stark shifted and broadened. The continuum spectrum also inherits an intricate pattern of resonances which reflect the competition between resonant scattering between the two atomic wells and between the linear potential and one or both atomic well(s). All results are analytic and can be easily plotted. The relation between the large orders of the divergent perturbative Stark shift series and the non-perturbative widths of quasi-bound levels is studied.

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Improving the accuracy of ab initio methods with summation approximants and singularity analysis

Goodson

2003

Int J of Quantum Chemistry

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ABSTRACT:The accuracy of Møller-Plesset (MP) perturbation theory and coupledcluster (CC) theory can be significantly improved, at essentially no increase in computational cost, by using summation approximants that model the way in which these theories converge to the full configuration interaction limit. Approximants for MP4 and CCSD(T) are presented, their size scaling is analyzed, and the functional analysis of the MP energy, on which the MP4 approximant is based, is discussed. The MP approximants are shown to have a form that is appropriate for describing resonance energies.

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Complex energies from real perturbation series for the LoSurdo-Stark effect in hydrogen by Borel-Padé approximants

Cited by 82 publications

References 31 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

Simple soluble molecular ionization model

Improving the accuracy of ab initio methods with summation approximants and singularity analysis

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