The Double Well Done Doubly-Well

Resurgence, Exact WKB, and Path Integrals in the Quantum Double Well

Introduction and Problem Statement

The symmetric quantum double-well potential is a paradigmatic example exhibiting the interplay between perturbative and non-perturbative phenomena in quantum mechanics. Its energy eigenvalues encode factorially divergent perturbative expansions and exponentially small splittings due to tunneling, phenomena unseen by perturbation theory at any finite order. Resurgence theory provides a formalism connecting these infractions of standard expansion, organizing the spectral data into a full trans-series. "The Double Well Done Doubly-Well" (2606.05282) presents a comprehensive, technical analysis of this problem, unifying the exact WKB method and the Euclidean path-integral approach. The paper further elucidates the geometric underpinnings of resurgence using elliptic curves and Lefschetz thimbles, and systematically pushes calculations to the four-instanton and three-loop regime.

Divergent Series and Trans-Series Structure

Perturbation theory near either well provides asymptotic expansions for the energies: $E_{N}(\hbar) = \sum_{k=1}^\infty e_k(N) \hbar^k$ with $e_k \sim k!$ for large $k$. The factorial divergence reflects the presence of instanton saddles in the path integral, leading to additional exponentially suppressed contributions. The total non-perturbative structure is a trans-series: $$E_N = \sum_k e_k \hbar^k + e^{-\frac{S_I}{\hbar}} \sum_k e_k^{(1)} \hbar^k + e^{-\frac{2S_I}{\hbar}} \sum_k (e_k^{(2)} + c_k^{(2)} \log \hbar) \hbar^k + \ldots$$ Here $S_I$ is the instanton action, and the coefficients across instanton sectors are algebraically related by resurgence.

The Borel transform constructs a resummation of the divergent perturbative series and reveals the analytic structure of the underlying physical function. Singularities in the Borel plane correspond directly to non-perturbative saddles in the semiclassical expansion. Resurgence theory, in the form developed by Écalle, enables the recursive determination of higher-instanton data from perturbative input via alien calculus.

Exact WKB Framework and Resurgent Constraints

The exact WKB method systematically regularizes the traditional WKB approximation by analytic continuation in the complex plane and Borel resummation, utilizing Voros symbols—exponentials of period integrals of the quantum momentum. Crucial is the treatment of analytic continuation across Stokes lines, formalized by Delabaere–Dillinger–Pham (DDP) relations, which relate discontinuities of Borel-resummed quantities to Stokes jumps associated with multi-instanton sectors.

The quantization condition for the spectrum involves both perturbative and instanton (non-perturbative) cycles: $1 + V_P(E) = \pm i \sqrt{V_N(E)}$ where $V_{P,N}$ are exponential period integrals over the respective cycles in the complexified phase space.

Figure 1: Complex $z$-plane with saddle points and associated Lefschetz thimbles illustrating the multi-saddle structure essential for resurgence.

The underlying geometry is that of an elliptic curve defined by the classical momentum, with the period integrals giving the actions of the relevant cycles. The period expansion enables computation to high order in $\hbar$ for multi-instanton sectors, relying on Picard–Fuchs relations between higher $k$ WKB integrals and the basic period data.

Resurgence is manifest in relations between the alien derivatives of Borel transforms (acting as 'bridges' between sectors). Explicit calculations verify the entanglement of the $e_k \sim k!$0-instanton and $e_k \sim k!$1-instanton sectors, and show that the reality of the energy spectrum emerges via intricate cancellation of imaginary parts at each instanton order, as quantified by the DDP and Écalle’s bridge equations.

Euclidean Path Integral and Lefschetz Thimbles

The path-integral approach decomposes the quantum partition function $e_k \sim k!$2 into sums over real and complex saddle points—determined by the finite-$e_k \sim k!$3 solutions to the Euclidean equations of motion, which are characterized by doubly-periodic Weierstrass elliptic functions. The actions are quantized according to their winding numbers around the fundamental cycles of the elliptic curve.

This saddle classification allows a Lefschetz thimble decomposition of the path integral: $e_k \sim k!$4 where $e_k \sim k!$5 are thimble contours and $e_k \sim k!$6 are intersection numbers—determined by the geometry of the complexified configuration space.

Figure 2: Four-sheeted Riemann surface associated with the 0D double-well model, revealing the multi-branched Borel structure reconstructed from the saddle analysis.

The functional determinant for quadratic fluctuations about each saddle is given in closed form in terms of period data. Collective coordinates (true zero modes) and quasi-zero modes (arising from nearly flat directions in multi-instanton configurations) are isolated and integrated over exactly, yielding precise $e_k \sim k!$7-dependence and demonstrating the polynomial structure of each sector in the large $e_k \sim k!$8 limit.

The path-integral framework provides a geometric realization of resurgence: the full spectrum emerges from the combined contributions of appropriately selected thimble integrals, with the properties of middle-dimensional integration cycles encoding the cancellation of Borel ambiguities.

Figure 3: Numerical computation of thimble structures for the Airy model, demonstrating analytic continuation around turning points and the geometry of Stokes phenomena relevant to the quantum double-well.

Numerical Results and Asymptotic Analysis

The analysis achieves explicit calculation of the spectrum up to four-instanton sectors and three-loop order in $e_k \sim k!$9, exposing the fine structure of the trans-series. The ground-state energy splitting and higher-level corrections are explicitly given and show exact agreement between the two methodological approaches.

Key numerical and analytical findings:

• The level splitting is exponentially small in $k$0:

$k$1

• All Borel ambiguities in perturbative and $k$2-instanton sectors cancel exactly against the contributions computed from lower sectors, as mandated by resurgence.
• Odd-instanton sectors affect only the even/odd splitting (parity dependence); even-instanton sectors yield common energy shifts.

Implications and Theoretical Outlook

This work establishes the symmetric double-well as a model laboratory where resurgence manifests in its most transparent, algebraically tractable form. The explicit parallel analysis using exact WKB and path-integral/Lefschetz-thimble formulations highlights the complementary roles of local and global geometry (period integrals, Riemann surfaces, and saddle-point network) in the analytic structure of quantum observables.

The techniques developed, especially the concrete construction of multi-instanton sectors using algebraic period data and their geometric interpretation via thimble integration, provide a rigorous methodology generalizable to more complex quantum systems, including quantum field theory and SUSY gauge quantum mechanics, where resurgence structures govern non-perturbative phenomena.

In AI and symbolic computation, the methods for automatic generation and manipulation of high-order perturbative and trans-series expansions—as well as the algebraic manipulation of period integrals—could enable new computational algebra systems, potentially automating aspects of multi-instanton calculus and the symbolic analysis of divergent series for a wide class of quantum systems.

Conclusion

This paper delivers a unified, technically precise, and deeply interconnected view of resurgence in the archetypal double-well system. By harmonizing exact WKB and path integral methods through their common elliptic geometry and resurgent algebra, it both clarifies historical confusions and provides explicit computational protocols for multi-instanton physics. The analysis paves the way for systematic, algebraic understanding of non-perturbative phenomena in quantum and statistical field theories grounded in the geometry of complex saddle networks and their associated trans-series structures (2606.05282).