Borel Summability Research Articles

Notes on Euler’swork on divergent factorial series and their associated continued fractions

Factorial series which diverge everywhere were first considered by Euler from the point of view of summing divergent series. He discovered a way to sum such series and was led to certain integrals and continued fractions. His method of summation was essentialy what we call Borel summation now. In this paper, we discuss these aspects of Euler’s work from the modern perspective.

A divergent generating function that can be summed and analysed analytically

We study a recurrence relation, originating in combinatorial problems, where the generating function, as a formal power series, satisfies a differential equation that can be solved in a suitable domain; this yields an analytic function in a domain, but the solution is singular at the origin and the generating function has radius of convergence 0. Nevertheless, the solution to the recurrence can be obtained from the analytic solution by finding an asymptotic series expansion. Conversely, the analytic solution can be obtained by summing the generating function by the Borel summation method. This is an explicit example, which we study detail, of a behaviour known to be typical for a large class of holonomic functions. We also express the solution using Bessel functions and Lommel polynomials.

Exact results for ionization of model atomic systems

We review recent rigorous results concerning the ionization of model quantum systems by time-periodic external fields. The systems we consider consist of a single particle (electron) with a reference Hamiltonian H0=−Δ+V0(x) (x∊Rd) having both bound and continuum states. Starting from an initially localized state ψ0(x)∊L2(Rd), the system is subjected for t≥0 to an arbitrary strength time-periodic potential V1(x,t)=V1(x,t+2π/ω). We prove that for a large class of V0(x) and V1(x,t), the wave function ψ(x,t) will delocalize as t→∞, i.e., the system will ionize. The only exceptions are cases where there are time-periodic bound states of the Floquet operator associated with H0+V1. These do occur (albeit rarely) when V1 is not small. For spatially rapidly decaying V0 and V1, ψ(x,t) is generally given, for very long times, by a power series in t−1/2 which we prove in some cases to be Borel summable. For the Coulomb potential V0(x)=−b|x|−1 in R3, we prove ionization for V1(x,t)=V1(|x|)sin(ωt−θ), V1(|x|)=0 for |x|>R and V1(x)>0 for |x|≤R. For this model, if ψ0 is compactly supported both in x and in angular momentum, L, we obtain that ψ(x,t)∼O(t−5/6) as t→∞.

On the Structure of Higher Order Simple-Pole Type Operators in Exact WKB Analysis

We introduce an appropriate class of higher order linear ordinary differential operators with a large parameter and with simple poles in their coefficients, and study their structure from the viewpoint of exact WKB analysis, i.e., the WKB analysis based on the Borel resummation. Our main result is that an operator in the class is expressed as a product of two operators Q and R, with Q being irrelevant to the Stokes geometry and with R being a second order simple-pole type operator studied by Koike ([Ko1], [Ko2], [Ko3]). This decomposition theorem gives us a connection formula for WKB solutions near the simple pole in question, and it is used to explain the background mechanism of some intriguing phenomenon we encounter in the study of a particular third order operator. Some discussions on the scope of the future development of the theory are included.

Résurgence-sommabilité de séries formelles ramifiées dépendant d’un paramètre et solutions d’équations différentielles linéaires

In this article, we investigate the Borel-resummable and resurgent properties of formal power series, solutions of a class of linear differential equations. In particular, we study the analytic dependence of the associated Borel sum in the parameters of the equations. We also analyse the whole resurgent structure associated with these formal power series thanks to the tool of microfunctions and the so-called alien derivatives. These analysis allow us to extend some results due to Y. Sibuya in [31], and to illustrate the powerful alien calculus of J. Ecalle.

Ramanujan summation and the exponential generating function $\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\zeta^{\prime}(-k)$

In the sixth chapter of his notebooks, Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions ∑n≥1log n enz and \(\sum_{n\geq 1}H_{n}^{(j)}~e^{-nz}\) where \(H_{n}^{(j)}=\sum_{m=1}^{n}\frac{1}{m^{j}}\) . We find a surprising relation between the two sums when j=1 from which follows a formula that connects the derivatives of the Riemann zeta-function at the negative integers to the Ramanujan sum of the divergent Euler sums ∑n≥1nkHn, k ≥ 0, where \(H_{n}=H_{n}^{(1)}\) . Further, we express our results on the Ramanujan summation in terms of the classical summation process called the Borel sum.

Gamow Vectors in a Periodically Perturbed Quantum System

Gamow vectors and resonances play an important role in scattering theory, especially in the physics of metastable states. We study Gamow vectors and resonances in a time-dependent setting using the Borel summation method. In particular, we analyze the behavior of the wave function ψ(x,t) for one dimensional time-dependent Hamiltonian \(H=-\partial_{x}^{2}\pm2\delta(x)(1+2r\cos\omega t)\) where ψ(x,0) is compactly supported.We show that ψ(x,t) has a Borel summable expansion containing finitely many terms of the form \(\sum_{n=-\infty}^{\infty}e^{i^{3/2}\sqrt{-\lambda_{k}+n\omega i}|x|}A_{k,n}e^{-\lambda_{k}t+n\omega it}\) , where λ k represents the associated resonance. This expression defines Gamow vectors and resonances in a rigorous and physically relevant way for all frequencies and amplitudes in a time-dependent model.For small amplitude (|r|≪1) there is one resonance for generic initial conditions. We calculate the position of the resonance and discuss its physical meaning as related to multiphoton ionization. We give qualitative theoretical results as well as numerical calculations in the general case.

Some Tauberian theorems for Borel summability methods

We investigate the conditions needed for a Borel summable sequence to be convergent. The results of this paper extend and improve the well-known result of Hardy and Littlewood (1913) [1].

Perturbation theory for metastable states of the Dirac equation with quadratic vector interaction

The spectral problem of the Dirac equation in an external quadratic vector potential is considered using the methods of the perturbation theory. The cases in one spatial dimension and in three spatial dimensions with spherical symmetry are explicitly studied. The problem is singular and the perturbation series is asymptotic, so that the methods for dealing with divergent series must be used. Among these, the distributional Borel sum appears to be the most well suited tool to give answers and to describe the spectral properties of the system. A detailed investigation is made in one and in three space dimensions with a central potential. We present numerical results for the Dirac equation in one space dimension: these are obtained by determining the perturbation expansion and using the Pad\'e approximants for calculating the distributional Borel transform. A complete agreement is found with previous nonperturbative results obtained by the numerical solution of the singular boundary value problem and the determination of the density of the states from the continuous spectrum.

Scaling behavior of three-dimensional group field theory

Group field theory is a generalization of matrix models, with triangulated pseudomanifolds as Feynman diagrams and state sum invariants as Feynman amplitudes. In this paper, we consider Boulatov's three-dimensional model and its Freidel–Louapre positive regularization (hereafter the BFL model) with an ‘ultraviolet’ cutoff, and study rigorously their scaling behavior in the large cutoff limit. We prove an optimal bound on large order Feynman amplitudes, which shows that the BFL model is perturbatively more divergent than the former. We then upgrade this result to the constructive level, using, in a self-contained way, the modern tools of constructive field theory: we construct the Borel sum of the BFL perturbative series via a convergent ‘cactus’ expansion, and establish the ‘ultraviolet’ scaling of its Borel radius. Our method shows how the ‘sum over triangulations’ in quantum gravity can be tamed rigorously, and paves the way for the renormalization program in group field theory.

Behavior of lacunary series at the natural boundary

We develop a local theory of lacunary Dirichlet series of the form ∑ k = 1 ∞ c k exp ( − z g ( k ) ) , Re ( z ) > 0 as z approaches the boundary i R , under the assumption g ′ → ∞ and further assumptions on c k . These series occur in many applications in Fourier analysis, infinite order differential operators, number theory and holomorphic dynamics among others. For relatively general series with c k = 1 , the case we primarily focus on, we obtain blow up rates in measure along the imaginary line and asymptotic information at z = 0 . When sufficient analyticity information on g exists, we obtain Borel summable expansions at points on the boundary, giving exact local description. Borel summability of the expansions provides property-preserving extensions beyond the barrier. The singular behavior has remarkable universality and self-similarity features. If g ( k ) = k b , c k = 1 , b = n or b = ( n + 1 ) / n , n ∈ N , behavior near the boundary is roughly of the standard form Re ( z ) − b ′ Q ( x ) where Q ( x ) = 1 / q if x = p / q ∈ Q and zero otherwise. The Bötcher map at infinity of polynomial iterations of the form x n + 1 = λ P ( x n ) , | λ | < λ 0 ( P ) , turns out to have uniformly convergent Fourier expansions in terms of simple lacunary series. For the quadratic map P ( x ) = x − x 2 , λ 0 = 1 , and the Julia set is the graph of this Fourier expansion in the main cardioid of the Mandelbrot set.

Short Time Existence and Borel Summability in the Navier–Stokes Equation in ℝ3

We consider the Navier–Stokes initial value problem, where 𝒫 is the Hodge-Projection to divergence free vector fields in the assumption that ‖f‖μ, β < ∞ and ‖ v 0 ‖μ+2, β < ∞ for β ≥ 0,μ > 3, where and [fcirc] (k) = ℱ [f (·)] (k) is the Fourier transform in x. By Borel summation methods we show that there exists a classical solution in the form t ∈ ℂ, , and we estimate α in terms of ‖ [vcirc]0 ‖μ+2, β and ‖ [fcirc] ‖μ, β. We show that ‖ [vcirc] (·; t) ‖μ+2, β < ∞. Existence and t-analyticity results are analogous to Sobolev spaces ones. An important feature of the present approach is that continuation of v beyond t = α−1 becomes a growth rate question of U(·, p) as p → ∞, U being is a known function. For now, our estimate is likely suboptimal. A second result is that we show Borel summability of v for v 0 and f analytic. In particular, Borel summability implies a the Gevrey-1 asymptotics result: , where , with A 0 and B 0 are given in terms of to v 0 and f and for small t, with ,

Infrared renormalons and single meson production in proton-proton collisions

In this article, we investigate the contribution of the higher-twist Feynman diagrams to the large-${p}_{T}$ inclusive pion production cross section in proton-proton collisions and present the general formulas for the higher-twist differential cross sections in the case of the running coupling and frozen coupling approaches. The structure of infrared renormalon singularities of the higher-twist subprocess cross section and the resummed expression (the Borel sum) for it are found. We compared the resummed higher-twist cross sections with the ones obtained in the framework of the frozen coupling approach and leading-twist cross section. We obtain, that ratio $R=({\ensuremath{\Sigma}}_{{\ensuremath{\pi}}^{+}}^{\mathrm{HT}}{)}^{\mathrm{res}}/({\ensuremath{\Sigma}}_{{\ensuremath{\pi}}^{+}}^{\mathrm{HT}}{)}^{0}$, for all values of the transverse momentum ${p}_{T}$ of the pion identically equivalent to ratio $r=({\ensuremath{\Delta}}_{\ensuremath{\pi}}^{\mathrm{HT}}{)}^{\mathrm{res}}/({\ensuremath{\Delta}}_{\ensuremath{\pi}}^{\mathrm{HT}}{)}^{0}$. It is shown that the resummed result depends on the choice of the meson wave functions used in calculation. Phenomenological effects of the obtained results are discussed.

Borel resummation of the ɛ-expansion of the dynamical exponent z in model a of the ϕ 4(O(n)) theory

We perform the Borel resummation of the currently known terms of the ɛ-expansion up to order ɛ 4 of the dynamical exponent z in the critical-behavior model A. We obtain the large-order asymptotic approximation of the ɛ-expansion of the dynamical exponent and find a significant discrepancy between the currently calculated orders of the expansion and the obtained asymptotic values. We discuss the influence of this deviation on the accuracy of the resummation results.

Formula omitted] and the τ hadronic width

Different choices exist for the renormalisation group resummation in the determination of $\alpha_s$ from hadronic $\tau$ decays: namely fixed-order (FOPT) and contour-improved perturbation theory (CIPT). The two approaches lead to systematic differences in the resulting $\alpha_s$. On the basis of a model for higher-order terms in the perturbative series, which incorporates well-known structure from renormalons, it is found that while CIPT is unable to account for the fully resummed series, FOPT smoothly approaches the Borel sum. Employing the model to determine $\alpha_s$ yields $\alpha_s(M_\tau)=0.316 \pm 0.006$, which after evolution leads to $\alpha_s(M_Z) = 0.1180 \pm 0.0008$.

Radiation of scalar oscillons in 2 and 3 dimensions

The radiation loss of small-amplitude radially symmetric oscillons (long-living, spatially localized, time-dependent solutions) in two- and three-dimensional scalar field theories is computed analytically in the small-amplitude expansion. The amplitude of the radiation is beyond all orders in perturbation theory and it is determined using matched asymptotic series expansions and Borel summation. The general results are illustrated on the case of the two- and three-dimensional sine-Gordon theory and a two-dimensional ϕ6 model. The analytic predictions are found to be in good agreement with the results of numerical simulations of oscillons.

Deep inelastic and dipole scattering on finite length hot [formula omitted] SYM matter

Deep inelastic scattering of R -currents and the scattering of a small dipole on finite length hot N = 4 SYM matter are discussed. In each case we find the scale when scattering becomes strong is determined by a saturation momentum Q s 2 ∼ L T 3 / x where L is the length of the matter. For R -currents we analyze the operator product expansion. For infinite length matter the series generated by the OPE is not Borel summable but we are able to determine the exponential part of the tunneling amplitude determining F 2 when Q 2 Q s 2 ≫ 1 from the position of the singularity closest to the origin on the real axis of the Borel plane. In finite length matter the OPE series is not convergent but it is Borel summable. When a small dipole, and the string connecting the ends of the dipole, pass through hot matter there is an induced motion of the string in the 5th dimension. When T 4 L cosh η , with the η the rapidity of the string, is large enough the string would normally break into several parts after leaving the medium, however, this cannot happen in the classical approximation in which we work.

Computation of the radiation amplitude of oscillons

The radiation loss of small amplitude oscillons (very long-living, spatially localized, time dependent solutions) in one dimensional scalar field theories is computed in the small-amplitude expansion analytically using matched asymptotic series expansions and Borel summation. The amplitude of the radiation is beyond all orders in perturbation theory and the method used has been developed by Segur and Kruskal in Phys. Rev. Lett. 58, 747 (1987). Our results are in good agreement with those of long time numerical simulations of oscillons.

An asymptotic functional-integral solution for the Schrödinger equation with polynomial potential

A functional integral representation for the weak solution of the Schrödinger equation with a polynomially growing potential is proposed in terms of an analytically continued Wiener integral. The asymptotic expansion in powers of the coupling constant λ of the matrix elements of the Schrödinger group is studied and its Borel summability is proved.

Ds(0±) Meson Spectroscopy in Gaussian Sum Rules

Masses of the Ds(0±) mesons are investigated from a view-point of ordinary light-heavy system in the framework of the Gaussian sum rules, which are worked out by means of the Laplacian transformation to the usual Borel sum rules. Using the standard input of QCD non-perturbative parameters, the corresponding mass spectra and couplings of the currents to the Ds(0±) mesons are obtained. Our results are mDs(0−) = 1.968 ± 0.016 ± 0.003 GeV and mmDs(0+) = 2.320 ± 0.014 ± 0.003 GeV, which are in good accordance with the experimental data, 1.969 GeV and 2.317 GeV.