Order-dependent Mapping Research Articles

From divergent perturbation theory to an exponentially convergent self-consistent expansion

For many nonlinear physical systems, approximate solutions are pursued by conventional perturbation theory in powers of the non-linear terms. Unfortunately, this often produces divergent asymptotic series, collectively dismissed by Abel as "an invention of the devil." An alternative method, the self-consistent expansion (SCE), has been introduced by Schwartz and Edwards. Its basic idea is a rescaling of the zeroth-order system around which the solution is expanded, to achieve optimal results. While low-order SCEs have been remarkably successful in describing the dynamics of non-equilibrium many-body systems (e.g., the Kardar-Parisi-Zhang equation), its convergence properties have not been elucidated before. To address this issue we apply this technique to the canonical partition function of the classical harmonic oscillator with a quartic $gx^{4}$ anharmonicity, for which perturbation theory's divergence is well-known. We obtain the $N$th order SCE for the partition function, which is rigorously found to converge exponentially fast in $N$, and uniformly in $g\ge0$. We use our results to elucidate the relation between the SCE and the class of approaches based on the so-called "order-dependent mapping." Moreover, we put the SCE to test against other methods that improve upon perturbation theory (Borel resummation, hyperasymptotics, Pad\'e approximants, and the Lanczos $\tau$-method), and find that it compares favorably with all of them for small $g$ and dominates over them for large $g$. The SCE is shown to successfully capture the correct partition function for the double-well potential case, where no perturbative expansion exists. Our treatment is generalized to the case of many oscillators, as well as to any nonlinearity of the form $g|x|^{q}$ with $q\ge0$ and complex $g$. These results allow us to treat the Airy function, and to see the fingerprints of Stokes lines in the SCE.

Order-dependent mappings: Strong-coupling behavior from weak-coupling expansions in non-Hermitian theories

A long time ago, it has been conjectured that a Hamiltonian with a potential of the form x2+ivx3, v real, has a real spectrum. This conjecture has been generalized to a class of the so-called PT symmetric Hamiltonians and some proofs have been given. Here, we show by numerical investigation that the divergent perturbation series can be summed efficiently by an order-dependent mapping (ODM) in the whole complex plane of the coupling parameter v2, and that some information about the location of level-crossing singularities can be obtained in this way. Furthermore, we discuss to which accuracy the strong-coupling limit can be obtained from the initially weak-coupling perturbative expansion, by the ODM summation method. The basic idea of the ODM summation method is the notion of order-dependent “local” disk of convergence and analytic continuation by an ODM of the domain of analyticity augmented by the local disk of convergence onto a circle. In the limit of vanishing local radius of convergence, which is the limit of high transformation order, convergence is demonstrated both by numerical evidence as well as by analytic estimates.

Summation of divergent series: Order-dependent mapping

Summation methods play a very important role in quantum field theory because all perturbation series are divergent and the expansion parameter is not always small. A number of methods have been tried in this context, most notably Padé approximants, Borel–Padé summation, Borel transformation with mapping, which we briefly describe and one on which we concentrate here, Order-Dependent Mapping (ODM). We recall the basis of the method, for a class of series we give intuitive arguments to explain its convergence and illustrate its properties by several simple examples. Since the method was proposed, some rigorous convergence proofs were given. The method has also found a number of applications and we shall list a few.

CONFORMAL MAPPING, POWER CORRECTIONS, AND THE QCD BOUND STATE SPECTRUM

We analyze the heavy quark bound state spectrum using an order-dependent conformal mapping to re-sum the peturbative expansion for current correlators. The procedure consists of two main steps. Firstly, the Borel plane structure of the truncated perturbative expression is modified to ensure consistency with the operator product expansion. This is analogous to a resummation of infrared renormalon chains. Secondly, this perturbative expansion is conformally mapped to a new series with improved convergence properties. This approach may be shown to induce power corrections consistent with existing condensates, and the resulting expansion may be ordered in powers of an infrared-analytic effective coupling. The technique is then applied to [Formula: see text] and [Formula: see text] sum rules without any explicit introduction of vacuum condensate parameters. Ground state masses for the vector, axial–vector and A′ channels are well reproduced, while results for the scalar–pseudoscalar mass splitting are less impressive.

The hydrogen atom in strong electric fields: summation of the weak field series expansion

The order dependent mapping method, whose convergence has recently been proven for the energy eigenvalue of the anharmonic oscillator, is applied to re-sum the standard perturbation series for the Stark effect of the hydrogen atom. We perform a numerical experiment up to the 50th order of the perturbation expansion. A simple mapping, suggested by the analytic structure and the strong field behavior, gives an excellent agreement with the exact value for an intermediate range of the electric field, 0.03 (a.u.) ≤ E ≤ 0.25 (a.u.). The imaginary part of the energy (the decay width) as well as the real part of the energy is reproduced from the standard perturbation series.

Further comments on the modified operator method

It is shown that a recently proposed perturbation approach to the two-step coherent-state approximation for one-dimensional oscillators leads to series which are asymptotically divergent, and coincident with the ones derived from the operator method. A simple method is provided to transform such a series into a convergent renormalized one making use of an order-dependent mapping. Results are excellent for all eigenvalues and all coupling constants.

Improved renormalized perturbation series

An improved version of a recently developed method for summing strongly divergent pertubation series is presented. The perturbation series is changed into a sequence of polynomials by means of a properly chosen order-dependent mapping. The limit of the resulting sequence is obtained approximately through the ϵ-algorithm of Shanks and Wynn. The Rayleigh-Schrödinger perturbation series for the eigenvalues of the anharmonic oscillator, the linear-confining potential and the Zeeman effect in hydrogen are discussed. Results for the strong-coupling limit show that the ϵ-algorithm accelerates the convergence of the renormalized perturbation series markedly.

Summation of strongly divergent perturbation series

A new method for summing strongly divergent perturbation series is presented. It is based on the change of the power series into a convergent sequence by means of an order-dependent mapping obtained from a simple scaling relation. The perturbation expansions for a one-dimensional integral and for the ground states of the anharmonic oscillator and of the linear confining potential model are accurately summed in the most unfavorable strong-coupling limit.

A method for summing strongly divergent perturbation series: The Zeeman effect in hydrogen

Abstract A new method for summing strongly divergent power series is presented. It consists of an order-dependent mapping that transforms the divergent series into a convergent sequence. The procedure is applied to the perturbation series of the Zeeman effect in hydrogen obtaining accurate energy values for the states 1s and 2p ±1 over a wide range of field strengths. Difficulties in handling the 2s state are pointed out.

The hydrogen atom in strong magnetic fields: Summation of the weak field series expansion

Using the weak field expansion, we have calculated the ground-state energy of the hydrogen atom in a magnetic field for values of the field up to about 10 13G. The perturbative expansion has been summed by an order-dependent mapping method. We compare our results with previous calculations. Our method allows us to obtain: first, more accurate values of the binding energy for a field up to 10 10G, then good results in the transition region between the two sets of accurate calculations of the literature, and finally still reasonably accurate values up to 10 13G.

Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory

We study numerically a method of summation of divergent series based on an order dependent mapping. We consider the example of a simple integral, of the ground-state energy of the anharmonic oscillator and of the critical exponents of φ34 field theory. In the case of the simple integral convergence can be rigorously proven, while in the other examples we can only give heuristic arguments to explain the properties of the transformed series. For the anharmonic oscillator we have compared our results to an accurate numerical solution (10−23) of the Schrödinger equation. For the critical exponents we have verified the consistency of our results with those obtained before from methods using a Borel transformation.