Introductory Lectures: Resurgence in Differential Equations, and Effective Summation Methods
Gerald Dunne
University of Connecticut
Isaac Newton Institute Spring School: Asymptotic Methods and Applications, March 22-26, 2021
Isaac Newton Institute Programme: Applicable Resurgent Asymptotics, 2021/2022 Basic Introduction to Resurgence: A Beginner’s Guide
Lecture 3 Basic Introduction to Resurgence: A Beginner’s Guide
1. Lecture 1: Resurgence & Linear Differential Equations I Trans-series and Stokes Phenomenon in ODEs I Borel Summation basics I Recovering Non-perturbative Connection Formulas 2. Lecture 2: Resurgence & Nonlinear Differential Equations I Nonlinear Stokes Phenomenon I Painlevé Equation examples I Parametric Resurgence & Phase Transitions 3. Lecture 3: Effective Summation Methods I Probing the Borel Plane Numerically I The Physics of Padé Approximation I Optimal Summation and Extrapolation Effective Summation Methods: A Basic Introduction
• computing perturbative coefficients is difficult • how do we make practical use of resurgence in a really difficult problem where we can only compute a small number of perturbative coefficients ? • consider a series which appears to be asymptotic, with generic leading large order behavior of the coefficients
Γ(a n + b) c ∼ S , n → ∞ n An
• A → location of the leading Borel singularity • b → nature of the leading Borel singularity • a → appropriate expansion variable • the Stokes constant S → normalization Effective Summation Methods: A Basic Introduction
PN cn • given just N terms of such a series n xn+1 (i) how can we sum accurately ? (ii) how can we extrapolate accurately to x = 0 ? (iii) how can we extrapolate into the complex x plane to probe Stokes transitions ? (iv) how can we extract non-perturbative exponentially small effects (connection) ? • resurgence suggests that local analysis of perturbation theory encodes global information • How much global information can be decoded from a FINITE number of perturbative coefficients ? • resurgent functions have orderly structure in Borel plane ⇒ develop extrapolation and summation methods that take advantage of this Effective Summation Methods: A Basic Introduction
• conclusion: it can make a BIG difference how we sum • processing the same perturbative input data in different ways can lead to vastly different levels of precision • the basic toolkit: ratio tests, series acceleration methods (e.g. Richardson), Padé approximants, orthogonal polynomials, Szegö asymptotics, continued fractions, conformal maps, uniformization maps, ... • the good news: many of these are actually very easy to implement → a simple set of exploratory procedures • the quality of the extrapolation of an asymptotic series is governed by the quality of the analytic continuation of the Borel transform Z ∞ f(x) = dt e−x tB[f](t) 0 • lesson 1: it is better to work in the Borel plane Resurgence in Nonlinear Differential Equations
• empirical observation: for “natural problems” the Borel plane often has structure Padé-Borel
• simple & powerful summation/extrapolation method
2N Z ∞ 2N X cn X cn = dt e−xt tn xn+1 n! n=0 0 n=0 Z ∞ −xt = dt e B2N [f](t) 0
• recall that the singularities of B2N [f](t) determine the non-perturbative physics
• but B2N [f](t) is a polynomial !
• as N → ∞, B2N [f](t) develops singularities • Padé is an excellent “low resolution” detector of singularity structures: “Padé-Borel” method Basics of Padé Approximation
• simple and efficient method to analytically continue a series beyond its radius of convergence P2N n • rational approximation to function F2N (t) = n cn t
2N RL(t) X P {F } (t) = = c tn + O t2N+1 [L,M] 2N S (t) n M n • completely algorithmic and algebraic (“built-in”)
• near-diagonal Padé: polynomials RN (t) & SN (t) satisfy the same 3-term recursion relation • hence a deep connection to orthogonal polynomials, and their asymptotics (Szegö ...) Basics of Padé Approximation
1. at very high orders Padé can be numerically unstable: a ratio of polynomials with very large coefficients. It is often more stable to convert to an ‘equivalent’ partial fraction
2N PN n N X n n an t X rn cnt ↔ ↔ PN n t − t n n dn t n n
2. truncated series → continued fraction, which often converges in all of C, minus a number of poles/cuts
2N 2N+1 1 2N+1 1+c1 t+···+c2N t +O t = +O t h1 t 1 + h2 t 1+ h t 1+ 3 .. 1 .+h2N t
3. near-diagonal Padé are related to continued fractions Basics of Padé Approximation
• a Padé approximant only has pole singularities. But for practical applications we are often interested in branch point singularities, e.g. Borel plane, critical exponents, ... • Padé represents a branch point as the accumulation point of an arc of poles: cf. potential theory (electrostatics) Im[t] Im[t]
10 10
5 5
Re[t] Re[t] -10 -5 5 10 -10 -5 5 10
-5 -5
-10 -10
1 1 1 1 Figure: Padé poles of 1 , 2πi 1 , and of 1 , 2πi 1 . (1−t) 5 (e 5 −t) 7 (1−t2) 5 (e 5 −t2) 7 Basics of Padé Approximation • symmetry of singularities is important for Padé Im[t] 2
1
Re[t] 5 10 15 -1
-2
Figure: Padé poles of 1/(1 − t + t2)1/5 • but the function has no singularities on the real t axis ! • Stahl (1997): in the limit N → ∞ Padé produces the “minimal capacitor” Im[t] 1.0
0.5
0.0 Re[t] 0.2 0.4 0.6 0.8 1.0
-0.5
-1.0
Figure: 1/t of Padé poles of 1/(1 − t + t2)1/5 Basics of Padé Approximation
1 • amazing sensitivity to exponent 2 1 1 inverse Padé poles of & 1 1 −10−6 (1 − t + t2) 2 (1 − t + t2) 2
Im[t] 1.0
0.5
0.0 Re[t] 0.2 0.4 0.6 0.8 1.0
-0.5
-1.0 • this sensitivity can be used to our advantage ... see later Basics of Padé Approximation
Exercise 3.1: Explore the Padé pole structures for various functions with interesting singularities. Basics of Padé Approximation
• Padé approximants are not very accurate near a branch point and branch cut, because of accumulating poles • Padé obscures repeated repeated singularities along the same line; but this is exactly what happens for resurgent Borel transforms in nonlinear problems • conformal mapping, before Padé, helps resolve these problems • make a conformal map into the unit disk, based on the leading Borel singularity/ies Conformal Mapping
• conformal map of cut domains: t → z Im[t] Im[t]
Re[t] Re[t]
Im[t] Im[z]
Re[t] Re[z] Padé-Conformal-Borel Method
• conformally map the Borel plane into the unit disk (based on the leading singularity), and then re-expand and do Padé inside the unit disk ⇒ significant improvement for resurgent Borel transforms • precision can be quantified (Szegö asymptotics) • much better near singularities: higher precision determination of Stokes constants and resolution of the location and nature of the singularity • better extrapolation from x = +∞ to x = 0 Basics of Padé Approximation