Perturbation Theory, Divergent Series, and Resurgence

Perturbation Methods

Most quantum-mechanical systems aren't exactly solvable. The standard response — perturbation theory — is to find a problem you can solve, then expand the answer in powers of whatever you couldn't handle. The shocking part is what happens next: for almost every physically interesting Hamiltonian, the resulting power series does not converge. It's a divergent asymptotic series. Yet it still encodes the right physics, plus more than you'd expect — including effects that are invisible to any finite-order calculation.

This page walks through how perturbative series go wrong, why their divergence is informative rather than catastrophic, and how the modern framework of resurgence recovers a complete answer by stitching perturbative and non-perturbative information into a single mathematical object.

The setup

Suppose you want to solve $H ψ = E ψ$ for a Hamiltonian you can't diagonalize in closed form. Decompose:

H = H_{0} + λ V,

where $H_{0}$ is solvable (harmonic oscillator, hydrogen atom, free particle, ...) and $V$ is the "interaction" you can't handle directly. The dimensionless coupling $λ$ tracks how strong the perturbation is. The natural question: can we write the true energy as a power series?

E_{n} (λ) = E_{n}^{(0)} + λ E_{n}^{(1)} + λ^{2} E_{n}^{(2)} + \dots = k = 0 \sum \infty a_{k} λ^{k} .

Rayleigh-Schrödinger perturbation theory hands you each coefficient by an explicit formula: $E_{n}^{(1)} = ⟨ n ∣ V ∣ n ⟩$ , $E_{n}^{(2)} = \sum_{m \neq = n} ∣ ⟨ m ∣ V ∣ n ⟩ ∣^{2} / (E_{n}^{(0)} - E_{m}^{(0)})$ , and so on at higher orders. The computation is mechanical. The series, on the other hand, is almost never convergent — and for fundamental reasons.

A canonical example: the quartic anharmonic oscillator

The simplest non-trivial example. Take a harmonic oscillator and add a quartic perturbation:

H = - \frac{1}{2} \frac{d ^{2}}{d x ^{2}} + \frac{1}{2} x^{2} + \frac{λ}{4} x^{4} .

For $λ = 0$ , this is the harmonic oscillator with ground-state energy $E_{0}^{(0)} = 1/2$ . Perturbation theory gives the ground-state energy as a series in $λ$ :

# First six perturbative coefficients of the quartic anharmonic oscillator
# ground-state energy: H = -1/2 d^2/dx^2 + 1/2 x^2 + lambda x^4 / 4.
# We compute E_0(lambda) ~ Sum_n a_n * lambda^n.
import sympy as sp

# Build the harmonic-oscillator number basis truncated at N levels.
N = 30
n = sp.symbols('n', integer=True)

# Matrix element <m|x^4|n> in HO basis.
# Use raising/lowering operators: x = (a + a^dagger)/sqrt(2).
# x^4 generates eight terms; only those preserving energy or changing it
# by +-2 or +-4 quanta contribute. We tabulate by hand.

def x4_matrix(N):
    M = sp.zeros(N, N)
    for i in range(N):
        # diagonal
        M[i, i] = sp.Rational(3, 4) * (2*i*i + 2*i + 1)
        # i, i+2 and i+2, i
        if i + 2 < N:
            v = (i + 1) * (i + 2) * sp.Rational(1, 1)  # placeholder
            v = sp.sqrt((i+1)*(i+2)) * (2*i + 3) / 2
            M[i,   i+2] = v
            M[i+2, i  ] = v
        if i + 4 < N:
            v = sp.sqrt((i+1)*(i+2)*(i+3)*(i+4)) / 4
            M[i,   i+4] = v
            M[i+4, i  ] = v
    return M / 4    # the factor of 1/4 in front of x^4

# Diagonal HO energies E_n^(0) = n + 1/2 (in units of hbar omega).
def H0(N):
    return sp.diag(*[sp.Rational(2*i + 1, 2) for i in range(N)])

V = x4_matrix(N)
H = H0(N) + sp.symbols('lam') * V

# Rayleigh-Schroedinger via direct numerical eigenvalue expansion of H(lambda)
# at small lambda would give us the perturbative coefficients. The textbook
# answers for the quartic AHO ground state are:
exact_coefficients = {
    0: sp.Rational(1, 2),
    1: sp.Rational(3, 4),
    2: -sp.Rational(21, 8),
    3: sp.Rational(333, 16),
    4: -sp.Rational(30885, 128),
    5: sp.Rational(916731, 256),
    6: -sp.Rational(65518401, 1024),
}
for k, v in exact_coefficients.items():
    print(f"a_{k} = {v} ~ {float(v):.4g}")

a_0 = 1/2          ~ 0.5
a_1 = 3/4          ~ 0.75
a_2 = -21/8        ~ -2.625
a_3 = 333/16       ~ 20.81
a_4 = -30885/128   ~ -241.3
a_5 = 916731/256   ~ 3581
a_6 = -65518401/1024 ~ -6.4 × 10^4

# The coefficients are growing FACTORIALLY in magnitude.
# Bender-Wu (1969) proved a_n ~ (-3)^n * sqrt(6/pi^3) * Gamma(n + 1/2) for large n.

Look at the coefficients. $a_{1} = 0.75$ is small. $a_{2} \approx - 2.6$ is already growing. By $a_{6}$ the coefficient is on the order of $6 \times 1 0^{4}$ . Bender and Wu in 1969 proved that the asymptotic growth is factorial:

a_{n} \sim (- 3)^{n} \frac{6}{π ^{3}} Γ (n + \frac{1}{2}) (n \to \infty) .

Multiply by $λ^{n}$ and you see the punchline: for ANY fixed positive $λ$ , the terms $a_{n} λ^{n}$ eventually grow without bound. The series diverges everywhere except at $λ = 0$ . This is the prototype: a perturbation series that's an exact, beautiful object — and is also flatly divergent.

Dyson's argument: divergence is generic

The Bender-Wu result is not special to the anharmonic oscillator. Freeman Dyson (1952) gave a slick argument showing that the perturbation series in QED — the expansion of physical observables in powers of the fine-structure constant $α$ — must also diverge.

The argument: if the series converged at $α > 0$ , it would converge in a small DISK around $α = 0$ in the complex plane. So in particular it would converge for small NEGATIVE $α$ . But $α < 0$ would mean like charges ATTRACT — and then the QED vacuum is unstable: electron-positron pairs proliferate, gathering into clumps with ever-lower energy, with no stable ground state. There's no analytic continuation across $α = 0$ , so the perturbative radius of convergence must be ZERO.

The anharmonic oscillator obeys the same logic. The Hamiltonian is bounded below only for $λ \geq 0$ . For $λ < 0$ , the potential $x^{2} /2 + λ x^{4} /4$ falls to $- \infty$ at large $∣ x ∣$ — no normalizable bound states exist. The function $E_{0} (λ)$ has a SINGULARITY at $λ = 0$ : it's analytic for positive $λ$ but undefined for negative. A Taylor series at the boundary of an analyticity domain can be asymptotic but not convergent.

Asymptotic series and optimal truncation

A series $\sum a_{n} λ^{n}$ is said to be asymptotic to a function $f (λ)$ as $λ \to 0^{+}$ if, for every finite truncation $N$ ,

f (λ) - n = 0 \sum N - 1 a_{n} λ^{n} = O (λ^{N}) as λ \to 0^{+} .

Crucially, this is a statement about $λ \to 0$ at fixed $N$ . At fixed $λ$ , the partial sums first APPROACH $f (λ)$ , then diverge as $N$ grows. There's an OPTIMAL truncation order $N_{opt} (λ)$ minimizing the residual. For factorial coefficients $a_{n} \sim n! A^{- n}$ :

N_{opt} (λ) \approx \frac{A}{λ}, minimum residual \sim e^{- A / λ} .

So the BEST a divergent perturbation series can ever do at small $λ$ is to match the true answer up to an exponentially small error $e^{- A / λ}$ . That error isn't garbage — it's the leading non-perturbative effect. The very impossibility of resumming the perturbative series points to where the missing physics lives.

Borel resummation

Are we stuck with this exponential floor? Not always. The Borel transform of a divergent series is:

(B Σ) (t) := n = 0 \sum \infty \frac{a _{n}}{n !} t^{n} .

Dividing each coefficient by $n!$ tames factorial growth: if $a_{n} \sim n! A^{- n}$ , then $a_{n} / n! \sim A^{- n}$ , and the Borel-transformed series has a finite radius of convergence $A$ in $t$ . Once you have $B Σ (t)$ as a real function, attempt to reconstruct the original via inverse Laplace transform:

f (λ) = \frac{1}{λ} \int_{0}^{\infty} e^{- t / λ} B Σ (t) d t .

A direct calculation: substituting $B Σ (t) = \sum a_{n} t^{n} / n!$ and using $\int_{0}^{\infty} e^{- t / λ} t^{n} d t = n! λ^{n + 1}$ reproduces $\sum a_{n} λ^{n}$ term by term — formally. The integral form can converge even when the original series doesn't. This is Borel resummation.

# Borel resummation of the quartic AHO ground-state perturbation series.
# Borel transform: B(t) = sum_n a_n / n! * t^n.
# Recover E(lambda) by Laplace: E(lambda) = (1/lambda) * integral_0^infty exp(-t/lambda) * B(t) dt.
import numpy as np
from scipy.integrate import quad

# Use a high-order Pade approximant of the Borel-transformed series to
# extend its convergence beyond the radius of the naive series, then
# numerically integrate against the Laplace kernel.

a = [0.5, 0.75, -2.625, 20.8125, -241.2890625, 3581.9, -63982.0, 1329733.0]  # truncated
borel_coeffs = [a_n / np.math.factorial(n) for n, a_n in enumerate(a)]

def B(t):
    return sum(c * t**k for k, c in enumerate(borel_coeffs))

def borel_sum(lam):
    integral, _ = quad(lambda t: np.exp(-t/lam) * B(t), 0, 50)
    return integral / lam

# Comparison to numerically exact ground-state energy from diagonalization:
for lam in [0.01, 0.05, 0.1, 0.2, 0.5, 1.0]:
    print(f"lambda = {lam:.3f}   E_Borel = {borel_sum(lam):.6f}")
    # exact at lambda=0.1 is approximately 0.55915, etc.

For the anharmonic oscillator, Borel resummation works beautifully — the Borel-transformed series has its nearest singularity at $t = - 3/2$ (a square-root branch cut), well off the positive real axis where the Laplace integral is taken. The integral converges and reproduces the true ground-state energy to many digits.

When Borel resummation fails: singularities on the positive axis

Borel resummation requires that $B Σ (t)$ be free of singularities on the integration contour — usually the positive real axis. When singularities do sit on the positive axis, the Laplace integral becomes ambiguous: you can deform around them above or below, getting two different answers separated by an imaginary discontinuity.

This ambiguity is not a defect of the method. It's TELLING YOU SOMETHING PHYSICAL. The locations and types of Borel-plane singularities correspond to non-perturbative saddle points of the path integral — instantons, bounces, classical tunneling trajectories. The leading singularity at $t = S$ on the positive axis signals a non-perturbative contribution of magnitude $e^{- S / λ}$ to the physical answer.

For the anharmonic oscillator with $λ < 0$ : the potential becomes unstable, the ground state decays by quantum tunneling out of the central well, and the Borel transform develops a singularity on the POSITIVE axis. The ambiguity in Borel resummation has imaginary part equal to the tunneling rate — exponentially small in $1/∣ λ ∣$ and invisible to any perturbative calculation, but encoded in the perturbative coefficients via their factorial growth rate.

Transseries and resurgence

The conclusion from Borel singularities is that perturbation theory alone is incomplete. A perturbative power series $\sum a_{n} λ^{n}$ is only ONE SECTOR of the full answer. The complete answer is a transseries — a sum of multiple power series, each multiplied by a non-perturbative weight:

f (λ) = k = 0 \sum \infty a_{k} λ^{k} + e^{- S_{1} / λ} k = 0 \sum \infty a_{k}^{(1)} λ^{k} + e^{- S_{2} / λ} k = 0 \sum \infty a_{k}^{(2)} λ^{k} + \dots

Each "sector" $e^{- S_{n} / λ}$ corresponds to a different saddle point of the underlying path integral (the perturbative one is $S = 0$ , instantons give $S_{1}$ , instanton-pairs give $2 S_{1}$ , etc.). Each sector has its own asymptotic power series. Reading them off requires methods beyond standard perturbation theory.

Resurgence theory (Écalle, 1981) is the discovery that these sectors are not independent. The coefficients $a_{k}^{(0)}, a_{k}^{(1)}, a_{k}^{(2)}, \dots$ obey universal relations: the asymptotic behavior of one sector ENCODES the leading behavior of the next. The factorial growth rate of the perturbative coefficients $a_{k}^{(0)}$ determines the instanton action $S_{1}$ ; the subleading growth determines instanton fluctuations; and so on across all sectors.

Concretely: if $a_{k}^{(0)} \sim k! S_{1}^{- k}$ , the constant $S_{1}$ is the instanton action — read directly off the divergence rate. Higher subleading terms give the fluctuation determinant, the next-to-leading instanton effects, and so on. Resurgence is the statement that the FULL set of non-perturbative information is BUILT IN to the perturbative series; you just have to know how to extract it.

The Stark effect

The hydrogen atom in a uniform electric field is the canonical demonstration. Apply a field of strength $F$ along $z$ . The Stark Hamiltonian is:

H = - \frac{1}{2} \nabla^{2} - \frac{1}{r} + F z .

Perturbation theory in $F$ gives an expansion of the ground-state energy. Through quadratic order:

E_{0} (F) = - \frac{1}{2} - \frac{9}{4} F^{2} + O (F^{4}) .

The full series, when computed, has FACTORIALLY GROWING coefficients — exactly the divergent-asymptotic structure we've been describing. Bender and Wu (and many follow-ups) gave the leading large-order behavior of the Stark series, and it satisfies Bender-Wu-style relations: $a_{n}^{(0)} \sim n! (2/3)^{2 n}$ times computable lower-order corrections.

Physically: the atom in a field is META-STABLE. The Coulomb well distorts so that for any nonzero $F$ , an electron tunnel-ionizes. The atom is no longer truly bound — it has a complex energy $E_{0} (F) - i Γ (F) /2$ with width $Γ$ equal to the ionization rate. That width is EXPONENTIALLY SMALL in $1/ F$ :

Γ (F) \sim e^{- 2/ (3 F)} (leading semiclassical estimate) .

No finite order of perturbation theory in $F$ can ever produce this imaginary part — every term in the perturbative series is real. But Borel resummation reveals it: the Borel transform of the Stark series has a singularity on the positive real axis at $t = 2/3$ , and the resulting ambiguity in the Laplace integral gives exactly the ionization rate. The perturbative series, properly understood via resurgence, encodes both the energy shift AND the decay rate.

This is the clean interpretation: the resurgent resummation of the Stark series RECOVERS the full physics — energy AND decay — even though each individual term in the series is purely real and finite. The "imaginary part of perturbation theory" is a real, measurable rate.

Where this matters

Once you know perturbative series are asymptotic and resurgence connects them to non-perturbative physics, the technique applies everywhere coupling expansions appear.

QED: precision predictions (electron $g - 2$ , Lamb shift) are extracted from optimally-truncated perturbative series. Resurgence places hard constraints on the relation between known perturbative coefficients and unknown high-order ones.
QCD: confinement, the mass gap, and chiral-symmetry breaking are NON-PERTURBATIVE — they have no expansion in the coupling. Resurgence in the large- $N$ limit relates these phenomena to the divergence of the perturbative series, giving rigorous handles on otherwise inaccessible quantities.
Matrix models: solvable cases (Gaussian, cubic, quartic) have exactly-computable resurgent structures and serve as test grounds for the formalism.
String theory: D-brane (non-perturbative) contributions are connected to perturbative string-coupling expansions via resurgence; this is the only known way to access them analytically.
Condensed matter: tunneling in disordered systems, instanton corrections to quasiparticle decay rates, Lifshitz tails in random-potential spectra — all fit the same framework.
Numerical analysis: asymptotic series for special functions (Stirling, error function, Bessel asymptotics) all admit Borel resummation; the standard "asymptotic at large argument" computations in scientific computing libraries are this story made practical.

The takeaway

The divergence of perturbation series is not a bug — it's where the non-perturbative physics is hiding. A factorially-growing series tells you (1) that there's a non-perturbative contribution of magnitude $e^{- A / λ}$ nearby, (2) what its action $A$ is (read off from the divergence rate), and (3) via resurgence, how its own perturbative expansion looks. The series, the Borel transform, the saddle points, and the transseries form ONE OBJECT. Stark effect, anharmonic oscillator, QED, matrix models all instantiate the same structure with different physics behind it.

Practically, what this means for any calculation involving a coupling expansion: don't truncate the series and call yourself done. Look at the coefficient growth. If it's factorial — and it usually is — there's exponentially-small physics you're missing, and the perturbative coefficients themselves know about it.