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arxiv: 2606.05282 · v1 · pith:AC7MK2XNnew · submitted 2026-06-03 · ✦ hep-th · hep-ph· quant-ph

The Double Well Done Doubly-Well

Aur\'elien Dersy , Matthew D. Schwartz This is my paper

Pith reviewed 2026-06-28 05:06 UTC · model grok-4.3

classification ✦ hep-th hep-phquant-ph
keywords resurgencedouble-well potentialtrans-seriesexact WKBpath integralinstantonsStokes phenomenaLefschetz thimbles
0
0 comments X

The pith

Resurgence organizes the exact spectrum of the symmetric double well into a single tightly-constrained trans-series.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the energy levels of the symmetric double-well potential mix perturbative expansions with exponentially small non-perturbative corrections that standard perturbation theory cannot see. It demonstrates that resurgence captures the full spectrum in one unified trans-series by performing explicit calculations to four-instanton order and three-loop accuracy. The authors develop two complementary methods in matching notation: exact WKB, where Stokes phenomena fix the series via quantization conditions, and the Euclidean path integral, where thimble integrals over quasi-zero modes determine the contributions. A reader would care because this provides a concrete, parameter-free example of how perturbative and non-perturbative physics become linked in a simple quantum system.

Core claim

The symmetric double well's energy spectrum forms a single resurgence trans-series. In exact WKB the Delabaere-Dillinger-Pham relations encode the Stokes phenomena that control analytic continuation of the wave function past turning points, and the resulting quantization condition in Voros symbols yields the complete trans-series. In the path integral the classical saddles are elliptic functions of Euclidean time, and a Lefschetz thimble decomposition on the quasi-zero-mode manifold selects the contributing saddles, producing a simpler trans-series whose T-dependence at each instanton order is a fixed polynomial.

What carries the argument

The resurgence trans-series for the energy levels, fixed by Delabaere-Dillinger-Pham relations in exact WKB and by Lefschetz thimble integrals over quasi-zero modes in the path integral.

If this is right

  • The quantization condition expressed in Voros symbols determines the entire trans-series from the local Stokes data alone.
  • At each instanton order the path-integral partition-function trans-series has a polynomial time dependence fixed solely by the thimble integrals.
  • The energy trans-series is more intricate than the partition-function trans-series, yet both are consistent through the computed orders.
  • The elliptic curve structure of the energy surface appears in both approaches but enters through different mechanisms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same local Stokes and thimble machinery may organize spectra for other potentials whose classical saddles form elliptic curves.
  • Extending the explicit calculations to five or six instantons would provide a direct test of whether the pattern continues without new parameters.
  • The wave-function trans-series itself, not only the energies, could be extracted from the same WKB data for comparison with numerical solutions.

Load-bearing premise

The local Delabaere-Dillinger-Pham relations together with the Lefschetz thimble decomposition on the quasi-zero-mode manifold suffice to fix the full trans-series without additional global information or fitting parameters.

What would settle it

An independent high-precision numerical computation of the energy levels that finds coefficients at four-instanton three-loop order differing from the predicted trans-series values would disprove the claimed organization.

Figures

Figures reproduced from arXiv: 2606.05282 by Aur\'elien Dersy, Matthew D. Schwartz.

Figure 1
Figure 1. Figure 1: The lobe area A has a trans-series expansion [20, 27] as a function of ε: A(ε) ∼ e − π 2ε  a0 + a1ε + a2ε 2 + · · · +  2ε π n n! + e − π 2ε  b0 + b1ε + · · ·   + · · · (2.2) The power series within each sector diverges factorially, with the large-order growth of an ∼ (2/π) n n!. The Borel transform of this series has its first singularity at |t| = π/2. This is the same as the distance to the nearest … view at source ↗
Figure 1
Figure 1. Figure 1: Phase portrait of the pendulum x¨ = sin x. The pendulum down at rest is x = π and the unstable equilibrium saddle is at x = 0 or x = 2π. (a) Unperturbed: the separatrix (blue) divides libration from rotation. The stable and unstable manifolds of each saddle point coincide exactly. (b) With rapid periodic forcing (ε > 0): the unstable manifold W u (solid red, departing each saddle) and stable manifold Ws (d… view at source ↗
Figure 2
Figure 2. Figure 2: Left: The complex z-plane with three saddle points: minima at z = ±1 and a maximum at z = 0. The original integration contour is the real line. Each thimble passes through its corresponding saddle point with matching color: J−1 (blue) from −∞ curving down to −i∞, J+1 (red) from +i∞ curving right to +∞, and J0 (green) along the imaginary axis. Right: The real part of the action S(z) = 1 8 (z 2 − 1)2 as a fu… view at source ↗
Figure 3
Figure 3. Figure 3: Four-sheeted Riemann surface for the 0D double well, defined by [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: The Airy function Ai(z) showing the transition from oscillatory behavior for z < 0 to exponential decay for z > 0. Right: Double rainbow with intensity computed from I(θ) ∝ | Ai[k 2/3 (θ − θR)]| 2 for each wavelength. The primary rainbow appears at ∼ 42◦ and secondary at ∼ 51◦ , with supernumerary bows visible as brightness variations inside each bow. and γA is the Airy integration contour2 that goes… view at source ↗
Figure 5
Figure 5. Figure 5: We can then see, for example, that for real [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: Top: Stokes diagrams showing the phase θ = arg z in each region. Bottom: The thimble structure computed numerically by steepest descent flow, with the Airy contour (dashed blue), J+ (green), and J− (orange). The asymptotically convergent regions are shaded in blue and denoted as A, B, C. regions. For the Airy integral, we sketch the Stokes lines as − + + (4.20) The solid lines are the Stokes lines at arg z… view at source ↗
Figure 6
Figure 6. Figure 6: Three-sheeted Riemann surface for the Airy function, showing [PITH_FULL_IMAGE:figures/full_fig_p031_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Image of the Airy contour (dashed blue) and the thimble contours [PITH_FULL_IMAGE:figures/full_fig_p033_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Stokes diagrams for the simple harmonic oscillator. Left: branch cuts directed to infinity. [PITH_FULL_IMAGE:figures/full_fig_p037_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Stokes graphs for the double well potential. The Stokes lines between [PITH_FULL_IMAGE:figures/full_fig_p040_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Left: oscillation of a particle with energy [PITH_FULL_IMAGE:figures/full_fig_p061_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Left: the inverted potential U(x) = − 1 8 (x 2 − 1)2 for the double well. Choosing a real energy in the range − 1 8 < ε < 0, the perturbative period ωP is imaginary and the non-perturbative period ωN is real. Im ωP and Re ωN are shown on the right. As ε → 0 we approach the instanton solution going between the two turning points so that the non-perturbative period gets large and ωP → −πi. so there is no si… view at source ↗
Figure 12
Figure 12. Figure 12: Exact instanton solutions from the Weierstrass [PITH_FULL_IMAGE:figures/full_fig_p065_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Exact (Weierstrass ℘) and approximate (tanh) 2-instanton solutions for the double well with T = 14. The instanton is centered at t1 = −T /4, the anti-instanton at t2 = T /4, with transition point t0 = 0. The action decomposes as S = Sb(∆1) +Sb(∆2) +Sb(∆3) +Sb(∆4) where each ∆ is the distance from an instanton center to the nearest boundary. approximate multi-instantons provide good approximations to the e… view at source ↗
Figure 14
Figure 14. Figure 14: Band structure of the Lamé equation as a function of the elliptic modulus [PITH_FULL_IMAGE:figures/full_fig_p075_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Starting from the exact n = 2 saddle at finite T, we follow gradient flow to the n = 0 saddle at x = −1. Left: action along the flow, plotted as S(α) with the normalization α(u=0) = T /2 and α(uend) = 0. Right: representative profiles x(t) along the same trajectory, labeled by α. The overall normalization of the flow parameter u is arbitrary. and a quasi-zero mode parameterized by α. We would like to para… view at source ↗
Figure 16
Figure 16. Figure 16: Lefschetz thimbles using the n = 2 effective action Seff from Eq. (5.170) with ℏ → ℏe iε . The perturbative saddles at x = ±1 are plotted at α = 0, T and shown as the orange dots, with their thimbles in orange. The (1, 0) real saddle is the black dot at α = T 2 . Its thimble goes off in the imaginary direction then veers real at the complex Stokes point associated with the (1, 1) saddle. The green thimble… view at source ↗
Figure 17
Figure 17. Figure 17: The n = 3 collective-coordinate geometry. On the left the effective action Seff(u, v) is shown. This action is valid near the n = 3 saddle, where αp = T /3 and u = v = 0, and descends to the n = 1 saddle where S = SI , approximated by the red plane. The right panel shows the real u, v domain with the n = 3 saddle at the center. The triangle edges at αp ≈ ln 6 represent the n = 1 instanton. The jagged line… view at source ↗
Figure 18
Figure 18. Figure 18: Downward-flow trajectories from the n = 4 saddle along the three unstable normal-mode directions x1, x2, and y at T = 40. The top row shows the normalized action S/SI along the flow coordinate, and the bottom row shows representative profiles x(t) (darker curves are earlier in the flow). The x1 and x2 modes each annihilate a single instanton–anti-instanton pair, reducing the action from 4SI to ≈ 2SI ; the… view at source ↗
read the original abstract

The symmetric double-well potential is one of the simplest quantum-mechanical systems in which perturbative and non-perturbative physics are deeply entangled. Its energy levels have non-analytic expansions in inverse powers of the inter-well separation, with factorially growing coefficients, while the parity splitting is exponentially small and invisible to perturbation theory. Resurgence ties the two features together, organizing the exact spectrum into a single tightly-constrained trans-series. This paper gives a self-contained account of this trans-series from two complementary approaches: exact WKB and the Euclidean path integral, developed in a common notation with explicit calculations through the four-instanton level and three-loop order. In exact WKB, Stokes phenomena encoded in the Delabaere--Dillinger--Pham relations control the analytic continuation of the wavefunction past turning points. The quantization condition expressed in terms of Voros symbols then determines the full trans-series. The DDP relations are local and do not require knowing the global topology of the energy surface, but that surface is an elliptic curve. In the path integral, elliptic curves enter differently: the classical saddle points are doubly-periodic elliptic functions of Euclidean time, and Stokes phenomena play out within the finite-dimensional manifold of quasi-zero modes rather than through analytic continuation of the wavefunction. A Lefschetz thimble decomposition determines which saddles contribute, and the resulting partition function trans-series is much simpler than the energy trans-series: at each instanton order the $T$-dependence is a polynomial fixed by the quasi-zero-mode thimble integrals. Together, the two approaches deploy a shared mathematical infrastructure in complementary ways, showing that the double well is an ideal setting to explore resurgence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper claims to provide a self-contained derivation of the trans-series for both the energy levels and the partition function of the symmetric double-well potential. It employs exact WKB with Delabaere-Dillinger-Pham (DDP) Stokes relations for the quantization condition in terms of Voros symbols, together with the Euclidean path integral using Lefschetz thimble decomposition over the quasi-zero-mode manifold, yielding explicit results through four-instanton order and three-loop order. The central assertion is that these local relations and thimble integrals determine all trans-series coefficients without global elliptic-curve periods or fitting parameters.

Significance. If the explicit calculations are correct, the work would be significant as a detailed benchmark for resurgence techniques in quantum mechanics, showing how two complementary formalisms (exact WKB and path integrals) produce consistent trans-series in a shared notation. The explicit four-instanton, three-loop results and the demonstration that local DDP relations suffice despite the elliptic-curve energy surface constitute a concrete strength; the paper ships explicit calculations at high perturbative and non-perturbative orders.

minor comments (2)
  1. The abstract is lengthy and contains several technical clauses; a shorter version focused on the main results and the parameter-free character of the construction would improve accessibility.
  2. The title is informal; consider whether a more descriptive title would better suit the journal's conventions while retaining the pun if desired.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the explicit four-instanton calculations, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

Derivation self-contained from exact WKB and path integrals; no reduction to inputs by construction

full rationale

The paper presents explicit four-instanton, three-loop trans-series for the double-well spectrum derived from Delabaere-Dillinger-Pham (DDP) Stokes relations in exact WKB and from Lefschetz thimble integrals over the quasi-zero-mode manifold in the path integral. The abstract and description state that DDP relations are local (do not require global elliptic-curve topology) and that thimble integrals fix the polynomial T-dependence at each order. No equations or claims indicate that coefficients are fitted to data, defined in terms of the output trans-series, or obtained via self-citation load-bearing steps. The construction is presented as parameter-free once the local relations and thimble decomposition are applied, consistent with the reader's assessment of score 2.0. No self-definitional, fitted-input, or ansatz-smuggling patterns appear in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the work rests on standard resurgence and elliptic-curve properties whose details are not visible here.

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