research31 min studyLesson 759 in Physics

Resurgence in quantum-mechanical examples

Fix a Borel convention, analyze controlled factorial-series examples, and explain how global spectral conditions enter quantum-mechanical transseries.

Before you begin

On this page

The question for today

What you’ll understand

Distinguish asymptotic truncation from convergence and Borel summation.
Compute a factorial-series remainder bound and a lateral ambiguity in explicit toy examples.
State which nonperturbative information requires global quantum-mechanical boundary conditions.

A divergent perturbation series can contain useful quantitative information, but its meaning depends on more than formally summing its terms. Factorial growth points toward singularities of a Borel transform and exponentially small scales. In selected quantum-mechanical problems, perturbative and instanton sectors fit into a transseries whose ambiguities cancel once the global spectral problem is fixed. We will first establish conventions using explicitly defined scalar integrals, then connect that controlled arithmetic to double-well quantum mechanics without claiming that every formal series uniquely specifies a nonperturbative theory.

An asymptotic series has an ordering prescription

For a formal series F of g with coefficients a n, define the Borel transform by dividing a n by n factorial. Recover a candidate sum through a Laplace integral with a prefactor one over g. This convention is important: an alternative transform for a series starting at g changes prefactors and residues. For positive g the positive real Borel axis is the default direction here, provided analytic continuation and growth allow that integral to exist.

Factorial coefficients typically make the original power series diverge at any fixed nonzero g, even though its first terms approximate a well-defined function. Truncating near the smallest term can be useful; adding arbitrarily many terms is not a convergence strategy. A Borel transform with singularities away from the chosen ray may be summable there. A singularity on the ray instead requires a lateral or other prescription and can produce an exponentially small ambiguity.

F(g)\sim\sum_{n=0}^{\infty}a_ng^n,\qquad \mathcal B F(t)=\sum_{n=0}^{\infty}\frac{a_n}{n!}t^n,\qquad \mathcal S F(g)=\frac1g\int_0^\infty dt\,e^{-t/g}\mathcal B F(t)

G and t are dimensionless in the examples. The final integral is a definition only when its analytic continuation and integration contour are specified and convergence holds. It does not assert that the original power series converges.↔ Scroll wide equations horizontally.

Two factorial toys expose singularity and contour effects

Define an exact positive-g function by the Borel integral of one over one plus t. Expanding the denominator gives alternating factorial coefficients. Its only finite Borel pole lies at negative one, away from the positive integration ray. A finite geometric-series remainder inside the integral bounds the asymptotic truncation error by the magnitude of the next factorial term. This is an elementary controlled example, not a calculation of a quantum double-well spectrum.

Changing to one over one minus t gives nonalternating factorial coefficients and a pole on the positive ray. Define the upper and lower lateral prescriptions explicitly; their difference is determined by a residue and scales as exponential minus one over g divided by g. That ambiguity cannot be represented by any finite power series at g = 0. Specifying only the coefficients leaves the choice between lateral sums unresolved unless further information defines the desired function.

\mathcal B F_{\mathrm{alt}}=\frac1{1+t},\qquad a_n=(-1)^n n!;\qquad \mathcal S_\pm F_{\mathrm{nonalt}}=\operatorname{PV}\frac1g\int_0^\infty\frac{e^{-t/g}}{1-t}\,dt\ \pm\frac{i\pi}{g}e^{-1/g}

The plus prescription passes above the pole and the minus prescription below it under the stated orientation from zero toward positive infinity. PV denotes principal value. Their difference is 2 pi i times e to minus one over g divided by g; this sign and power depend on the declared Borel convention.↔ Scroll wide equations horizontally.

A quantum spectrum needs global information

For double-well and periodic quantum-mechanical potentials, an expansion around one harmonic minimum misses tunneling effects visible in level splittings or band structure. Semiclassical sectors include exponentials of instanton actions, fluctuation series around those sectors, and sometimes logarithms from correlated multi-event or nearly flat directions. The term transseries refers to that broader structure. Cancellation of imaginary lateral ambiguities between sectors can be demonstrated in specified examples using compatible conventions.

A local perturbative calculation must still be combined with the global eigenvalue condition, such as normalizability or periodic and Bloch boundary conditions. Those conditions select the physical level or band and fix information not supplied by an arbitrary formal coefficient list alone. Resurgent relations in controlled quantum-mechanical systems are powerful evidence and computational tools; they do not by themselves furnish a convergent universal algorithm for all strongly coupled field theories. One should distinguish an exact integral identity, an asymptotic relation, and a model-dependent semiclassical construction.

Work it through

Bound an alternating factorial truncation

For the exactly defined positive-g integral $F(g)=g^{-1}\int_0^\infty e^{-t/g}(1+t)^{-1}dt$ , use g = 0.1. Compute the expansion through n = 3 and bound the remainder with its sign using the finite geometric identity.

The first four terms are $1-0.1+2(0.1)^2-6(0.1)^3=1-0.1+0.02-0.006=0.914$ . This is a truncated asymptotic expansion.
The finite denominator identity has remainder $t^4/(1+t)$ after these terms. Its Laplace integral is positive and bounded above by the integral of t to the fourth, because $0<(1+t)^{-1}\le1$ on this ray.
The upper remainder bound is $4!(0.1)^4=0.0024$ . Hence $0.914<F(0.1)<0.9164$ . The result is a bound for this exact integral, not a claim that the infinite factorial series converges.

The result

The n = 3 truncation is 0.914, with a positive remainder below 0.0024. The exact defined function therefore lies between 0.914 and 0.9164.

The living notebook

Explore the model

Interactive lab

Interactive lab · spectral

More terms require a convergence claim

Reset, set Mode count N = 4 and Coefficient decay p = 2, and predict smallest wavelength 2π/4 and highest-mode coefficient 1/16. For a fresh comparison Reset, set N = 12 and p = 2, and predict wavelength 2π/12 and coefficient 1/144. Inspect the initial paused curves and the selected-mode readouts.

Finite sumHighest included mode

Sum at x = π/2: 0.867
Smallest wavelength: 2π / 5
Highest-mode coefficient: 0.2

Mode count N5Coefficient decay p1

t = 0 s

A finite sum of sin(nx − t)/nᵖ. Playback changes t; increasing the cutoff includes finer spatial scales. This is not a proof about a continuum limit or renormalization. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThis instrument sums sin(n x − t)/n to the power p. Its coefficients are not factorial, and it performs no Borel continuation, lateral integration, instanton calculation, or quantum spectral resummation. It illustrates only that a truncation is a chosen finite expression.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Normalizability and other global spectral conditions select the physical problem. Resurgent relations connect sectors in controlled examples, but an arbitrary local formal series alone does not specify every boundary condition or nonperturbative choice.

Your turn

Try it yourself

Use the nonalternating toy with Borel transform $1/(1-t)$ and g = 1/5 under the displayed lateral convention. Find each lateral sum’s imaginary part and their difference. Does an ambiguity of that size prove that an exact stable quantum energy is complex?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Use the residue term and keep the one-over-g prefactor in this convention.

Compare with the solution

The imaginary parts are $\pm5\pi e^{-5}$ , and the upper-minus-lower difference is $10\pi i e^{-5}$ . This is an ambiguity of the specified isolated Borel prescription. It does not prove that a stable self-adjoint quantum Hamiltonian has complex energy. In an appropriate quantum-mechanical transseries, other sectors and the physical global condition can cancel such prescription dependence; that requires a specified model and consistent matching.

Keep the idea

Take it with you

A Borel convention fixes factorial normalization, Laplace prefactors, and contour signs.
Asymptotic truncation and lateral ambiguities can be analyzed exactly in controlled toys.
Quantum-mechanical resurgence includes global spectral conditions and does not automatically solve every field theory.

Read further

Original sources and deeper explanations to follow at your own pace.

Dunne and Ünsal, Uniform WKB, Multi-instantons, and Resurgent Trans-Series, sections I–III (double wells, global quantization, and ambiguity cancellation)

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