Resonances in Stark effect and perturbation theory

Graffi, Sandro; Grecchi, V.

doi:10.1007/bf01940333

Cited by 101 publications

(43 citation statements)

References 18 publications

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“…(346) and (347) and the operator H(F ) for complex values of F is the interpretation of (347) as the analytic continuation of (346) from F > 0 to −F = e iπ F . Remark 3 is proven in [322].…”

Section: B2 Distributional Borel Summability Of the Stark Effectmentioning

confidence: 89%

“…The first proof goes back to Titchmarsh, and relies on the separability in squared parabolic coordinates. A more recent and direct proof can be found in [322].…”

Section: B2 Distributional Borel Summability Of the Stark Effectmentioning

confidence: 90%

“…Assertions 1, 2, 3 and 4 of the theorem have been proven in [322] through the separability of the problem in squared parabolic coordinates recalled above, and in [326] by a direct analysis of the operator H(F, θ). The complete equivalence between the two methods is proven in [206].…”

Section: B2 Distributional Borel Summability Of the Stark Effectmentioning

confidence: 90%

“…dilated operators H(θ) and H F (θ) has made it possible to obtain a rigorous existence proof in the physically most relevant cases (atomic systems, Stark effect [321,322]). …”

Section: A2 Analytic Continuation Of the Resolvent And Resonancesmentioning

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: B2 Distributional Borel Summability Of the Stark Effectmentioning

confidence: 89%

“…The first proof goes back to Titchmarsh, and relies on the separability in squared parabolic coordinates. A more recent and direct proof can be found in [322].…”

Section: B2 Distributional Borel Summability Of the Stark Effectmentioning

confidence: 90%

Section: B2 Distributional Borel Summability Of the Stark Effectmentioning

confidence: 90%

“…dilated operators H(θ) and H F (θ) has made it possible to obtain a rigorous existence proof in the physically most relevant cases (atomic systems, Stark effect [321,322]). …”

Section: A2 Analytic Continuation Of the Resolvent And Resonancesmentioning

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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“…Rather mathematically motivated investigations regarding the Borel summability of the divergent perturbation series for the LoSurdo-Stark effect were performed in [12] and [47], and it was established that the perturbation series of the LoSurdo-Stark effect is Borel summable in the distributional sense (for the definition of "distributional Borel summability" we refer to [48]). Here, to supplement the mathematically motivated investigations, we consider the calculation of transforms of the divergent series, which use as input data only a finite number of perturbative coefficients and exhibit apparent convergence to the complete, physically relevant result.…”

Section: Introductionmentioning

confidence: 99%

Resummation of the divergent perturbation series for a hydrogen atom in an electric field

Jentschura

2001

Phys. Rev. A

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We consider the resummation of the perturbation series describing the energy displacement of a hydrogenic bound state in an electric field (known as the Stark effect or the LoSurdo-Stark effect), which constitutes a divergent formal power series in the electric field strength. The perturbation series exhibits a rich singularity structure in the Borel plane. Resummation methods are presented which appear to lead to consistent results even in problematic cases where isolated singularities or branch cuts are present on the positive and negative real axis in the Borel plane. Two resummation prescriptions are compared: (i) a variant of the Borel-Padé resummation method, with an additional improvement due to utilization of the leading renormalon poles (for a comprehensive discussion of renormalons see [M. Beneke, Phys. Rep. 317, 1 (1999)]), and (ii) a contour-improved combination of the Borel method with an analytic continuation by conformal mapping, and Padé approximations in the conformal variable. The singularity structure in the case of the LoSurdo-Stark effect in the complex Borel plane is shown to be similar to (divergent) perturbative expansions in quantum chromodynamics.11.15. Bt, 11.10.Jj, 32.60.+i, 32.70.Jz

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Hydrogen Supersymmetry: A New Method in the Perturbation Theory

Muljarov

2002

phys. stat. sol. (b)

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The O(4) supersymmetry of the hydrogen atom is utilized to construct a complete basis using only the bound state wave functions. For a large class of perturbations, an expansion of the electron (exciton) wave function into such a complete set reduces the perturbed Schrö dinger equation to the ordinary eigenvalue problem with equidistant unperturbed energy spectrum. A high-order polarizability of the ground state of the hydrogen atom in a static electric field is calculated via the Rayleigh-Schrö dinger perturbation theory for illustration of the new method.

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Resonances in Stark effect and perturbation theory

Cited by 101 publications

References 18 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

Resummation of the divergent perturbation series for a hydrogen atom in an electric field

Hydrogen Supersymmetry: A New Method in the Perturbation Theory

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