Construction of the Strong Coupling Expansion for the Ground State Energy of the Quartic, Sextic, and Octic Anharmonic Oscillator via a Renormalized Strong Coupling Expansion

Asymptotic response of observables from divergent weak-coupling expansions: a fractional-calculus-assisted Padé technique

Appropriate constructions of Padé approximants are believed to provide reasonable estimates of the asymptotic (large-coupling) amplitude and exponent of an observable, given its weak-coupling expansion to some desired order. In many instances, however, sequences of such approximants are seen to converge very poorly. We outline here a strategy that exploits the idea of fractional calculus to considerably improve the convergence behavior. Pilot calculations on the ground-state perturbative energy series of quartic, sextic, and octic anharmonic oscillators reveal clearly the worth of our endeavor.

Free-space asymptotic far-field series

A systematic procedure for deriving the complete asymptotic series in inverse powers of the distance from the origin of free-space 2D and 3D, scalar and vectorial, monochromatic electromagnetic fields is derived here. Each term of the series is expressed in closed form through the use of a differential operator acting on the angular spectrum. A simple recursive routine for computing the derivatives is provided. Examples of application are also given.

Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals

When the Weniger transformation is systematically used in resumming the asymptotic series arising from the application of the steepest descent method can display dramatic numerical instabilities that prevent it from improving the accuracy achievable via superasymptotics. In the present paper an explanation of such pathologies through the concept of resurgence, introduced by Berry and Howls several years ago within the context of hyperasymptotics, is proposed. In particular, the way the topology of the whole set of the saddles influences the resummation capabilities of the Weniger transformation is here investigated for the integrals defining the lowest-order cuspoid diffraction catastrophes. Eventually, a powerful and easily implementable resummation scheme, based on a joint use of the Weniger transformation and hyperasymptotics, is proposed for taking care of the above pathologies. Such a joint action seems to encompass the main virtues of both approaches, and the preliminary numerical results obtained from its application show that relative errors several orders of magnitude smaller than those achievable via superasymptotics can be achieved with modest implementation efforts.

Evaluation of diffraction catastrophes by using Weniger transformation

Weniger transformation is a powerful nonlinear sequence transformation that, when applied to the sequence of the partial sums of a divergent or a slowly convergent series, can convert it to a fast-converging sequence. Weniger transformation is not yet well known in optics. Diffraction catastrophes are fundamental tools for evaluating an optical field in proximity to caustics and singularities. The action of the Weniger transformation on the power series representation of diffraction catastrophes is numerically studied for two particular cases, corresponding to the Airy and the Pearcey functions. The obtained results clearly show that Weniger transformation could become a computational tool of great importance for summing several types of series expansions in optics.

Summation of perturbation series of eigenvalues and eigenfunctions of anharmonic oscillators

A perturbation approach to compute the bound states of the Schrödinger equation HPsi=EPsi with H0+lambdaV and Psi(x=+/- infinity)=0 is studied. The approach involves solving the corresponding Dirichlet problem H(R)Psi(R)=E(R)Psi(R) on a finite interval [-R,R] by the Rayleigh-Schrödinger perturbation theory (RSPT). The method is based on the fact that E(R),Psi(R) converge to E,Psi as R--> infinity. The model problems to study the summability properties of the RSPT series E(R)= sum(infinity)(k=0)E((k))(R)lambda(k) are the anharmonic oscillators H=p(2)+x(2)+lambda(x)(2M), with M=2,3,4 for which the RSPT produces strongly divergent series E= sum(infinity)(k=0)E((k))lambda(k). The summation of the latter series with large lambda for the octic case is considered as an extremely challenging summation problem, in part, since it was rigorously proven that the Padé approximants cannot converge and the two-point Padé approximants, which combine information of the renormalized weak coupling and strong coupling expansions, give relatively good results. The calculations of this work show that the ordinary Padé approximants from the sole un-normalized E(R) series for the octic oscillator give accurate results with small or large lambda. The coefficients E((k))(R) are calculated with the eigenvalue series of an operator H(Rn), whose resolvent converges to that of H(R) as n--> infinity. The Padé approximants of the RSPT eigenfunction series Psi(R)= sum(infinity)(k=0)psi((k))(R)lambda(k) also provide accurate results for the octic oscillator.

Summing Lax series for nonparaxial beam propagation

It is shown that the approach proposed by Lax et al. [Phys. Rev. A 11, 1365 (1975)] for studying the propagation of an electromagnetic beam beyond the paraxial approximation can be efficiently employed evenwhen the beam under consideration presents a very nonparaxial character. The method that we present consists of applying a nonlinear resummation scheme, the so-called delta transformation, to the divergent perturbative series arising from the Lax scheme. Numerical results pertinent to the evaluation of transverse and longitudinal components of the electric field are presented for the particular case of vectorial Gaussian beams, showing the effectiveness of the method.

Resummation of QED perturbation series by sequence transformations and the prediction of perturbative coefficients

We propose a method for the resummation of divergent perturbative expansions in quantum electrodynamics and related field theories. The method is based on a nonlinear sequence transformation and uses as input data only the numerical values of a finite number of perturbative coefficients. The results obtained in this way are for alternating series superior to those obtained using Pade approximants. The nonlinear sequence transformation fulfills an accuracy-through-order relation and can be used to predict perturbative coefficients. In many cases, these predictions are closer to available analytic results than predictions obtained using the Pade method.