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arXiv:2606.05282

The Double Well Done Doubly-Well

Aurélien Dersy, Matthew D. Schwartz

hep-thhep-phquant-phSubmitted Jun 3, 2026• Announced Jun 5, 2026
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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.

Comments
125+28 pages, 18 figures

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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 TTT-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.
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We study non-Hermitian quantum mechanics of an inverted triple-well potential within the exact WKB framework. For a single classical potential, different Siegert boundary conditions define three distinct quantum problems: the PT-symmetric, resonance, and anti-resonance systems. For each case, we derive the exact quantization condition and construct the associated trans-series solution. By identifying the resurgent structures and cancellations in these non-Hermitian setups, we obtain the median-summed series, clarifying when the spectra are real or complex in accordance with the physical properties of each system. Establishing explicit links to the semi-classical path integral formalism, we elucidate the roles of bounce and bion configurations in these non-Hermitian systems. This analysis predicts PT-symmetry breaking, which we also verify numerically. Using the median quantization conditions, we prove the existence of this symmetry breaking and establish an exact equation for the exceptional point, which emerges as a remarkably simple algebraic relation between the bounce and bion actions. We further show that the median-summed non-perturbative correction to the spectrum vanishes at the exceptional point, while the resurgent structure survives through a universal minimal trans-series. For the resonance and anti-resonance systems, we find that the exact median-summed spectra are related by complex conjugation, representing time reversal in this setting, are necessarily complex, and do not exhibit an exceptional point. Although their spectra differ significantly from the PT-symmetric case, they share the same minimal trans-series. By maintaining explicit links with the path integral saddles and the formal theory of resurgence, our analysis provides a unified and general perspective on the quantization of non-Hermitian theories.
We study the relation between the exact-WKB Delabaere--Dillinger--Pham Stokes jumps and the multi-instanton expansion of a one-dimensional potential with NNN equal-depth minima. For a single barrier this relation is the Bogomolny--Zinn-Justin cancellation between the instanton--anti-instanton ambiguity and the Borel ambiguity of perturbation theory, and we extend it to every consecutive barrier interval and to the integration contours themselves. In the dilute large-TTT regime at fixed nonperturbative grade, we derive the quasi-zero mode integral over the instanton separations from the Euclidean path integral and reduce it to Gamma and Hankel periods. With the local Weber level function of each well and one normalization per well fixed by the local Weber connection matrix, these periods reproduce the two lateral connected exact-WKB coefficients. The two lateral contours of each instanton--anti-instanton separation differ by exactly one Hankel cycle, and this linear Picard--Lefschetz relation reproduces the Stokes jump of the connected coefficient. A numerical solution of the Schrödinger equation for a symmetric triple well confirms the resulting two-barrier level splitting. At higher fixed grade, the map from ordered event sequences to normalized scalar coefficient products is not one-to-one. At leading dilute order the scalar weight of a closed event sequence depends only on the numbers of barrier crossings, IIˉ\mathcal I\bar{\mathcal I}IIˉ pairs, and II\mathcal I\mathcal III/IˉIˉ\bar{\mathcal I}\bar{\mathcal I}IˉIˉ pairs. We construct two inequivalent grade-six triple-well event sequences with the same product of local Weber amplitudes, which embed in every potential with N≥3N\ge3N≥3 wells. Our results hold within the regular dilute, one-loop, local Weber approximation.
The ground-state energy splitting due to tunneling in two-dimensional double wells of the form $V(x,y)=\frac{{({x}^{2}\ensuremath{-}{R}^{2})}^{2}}{8{R}^{2}}+\frac{{x}^{2}\ensuremath{-}{R}^{2}}{{R}^{2}}\ensuremath{\gamma}y+\frac{{\ensuremath{\omega}}^{2}}{2}{y}^{2}$ is calculated. Several results are reported. First, we give a systematic WKB expansion of the splitting in series in powers of ${R}^{\ensuremath{-}2}$, each term of the series being a finite polynomial in ${\ensuremath{\gamma}}^{2}$. We find an ascending sequence of the values of the parameter $\ensuremath{\gamma}$ characterizing the curvature of the classical path, for which the successive corrections to the leading order vanish. This effect arises because curvature of the path and quantum nature of motion cancel each other; it does not appear for one-dimensional double wells. Second, we find that for large curvatures, such as for those describing the proton transfer in a malonaldehyde and hydroxalate anion, this expansion is of no practical use. Thus, the WKB expansion is reordered to a strong coupling form, each term of the series in powers of ${R}^{\ensuremath{-}2}$ being an infinite series in powers of ${\overline{\ensuremath{\gamma}}}^{2}$, $\overline{\ensuremath{\gamma}}=\ensuremath{\gamma}/R$. Third, we find that the radius of convergence of the series is determined by the singularity at ${\overline{\ensuremath{\gamma}}}_{s}=\ensuremath{\omega}/2$. At the singularity the system changes its character from being a double well to become a single well. Close to this singularity the classical action and its first quantum correction are found to be nonanalytic functions of $\overline{\ensuremath{\gamma}}$, most likely of the form ${[1\ensuremath{-}{(\overline{\ensuremath{\gamma}}/{\overline{\ensuremath{\gamma}}}_{s})}^{2}]}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=1/2$ and $\ensuremath{\alpha}=\ensuremath{-}1/2$ for the classical action and its first quantum correction, respectively. Since in the semiclassical regime of large RRR the splitting is exponentially dependent on the value of the classical action and its first quantum correction, close to the singularity we establish strong sensitivity of the splitting on slight variations of the parameter $\overline{\ensuremath{\gamma}}$ entering the Hamiltonian linearly.
In this work, we study the resurgence program in holomorphic quantum mechanics. As a specific problem, we investigate the resurgence in the quartic anharmonic oscillator within holomorphic quantum mechanics, using the Bargmann representation of bosonic operators. In this framework, the perturbative energy series is shown to be Gevrey-1 and Borel summable only after continuation across the Stokes line. The instanton operator, realized as a coherent-state displacement in the Segal--Bargmann space, provides an explicit operatorial bridge between perturbative coefficients and non-perturbative sectors. Alien derivative relations generate the full resurgence triangle characteristic of the Bender--Wu model, and the resummed energy is expressed as a trans-series via a ratio of expectation values involving this instanton operator. As a concrete demonstration, we compute the first seven energy levels (n=0,…,6n=0,\dots,6n=0,…,6) up to sixth order in the coupling ggg; the resulting exact rational coefficients reproduce the classic Bender--Wu results, confirming the consistency and power of the holomorphic resurgence approach.
An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency ω and the difference of the minima is close to a multiple of �ω. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only reproduces the instanton result for a symmetric potential, but also elucidates the appearance of resonances of tunneling in the asymmetric potential. Key words: quantum tunneling, energy splitting, double-well potential PACS: 03.65.Sq, 03.65.Ge, 82.20.Xr 1