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Multi-Instantons and Exact-WKB Stokes Jumps in Multi-Well Quantum Mechanics

This paper extends the Bogomolny–Zinn-Justin cancellation mechanism to multi-well quantum systems by deriving multi-instanton contributions from Euclidean path integrals, demonstrating how Picard–Lefschetz relations between integration contours reproduce exact-WKB Stokes jumps, and identifying ambiguities in event sequence mappings at higher nonperturbative orders.

Original authors: Alireza Behtash, Syo Kamata

Published 2026-10-15

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CC BY 4.0

Original authors: Alireza Behtash, Syo Kamata

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of quantum mechanics, particles do not behave like tiny billiard balls rolling along a track. Instead, they exist as clouds of probability, capable of slipping through walls that should be impenetrable. This phenomenon, known as tunneling, is the reason the sun shines and the reason our computers can store data. For decades, physicists have used two different mathematical languages to describe how these particles move between the deep valleys of a potential energy landscape. One language, called exact-WKB, treats the particle's path as a wave that interferes with itself, creating a complex pattern of jumps and shifts. The other language, the path integral, views the particle as a traveler taking every possible route at once, summing up the contributions of countless journeys. While these two approaches describe the same physical reality, they have often seemed like two different maps of the same territory, making it difficult to see how the specific steps of one map translate into the features of the other.

A team of researchers has now bridged this gap for a specific class of problems involving particles moving in a landscape with multiple equal-depth valleys. They focused on a scenario where a particle tunnels back and forth between these valleys, a process that creates a delicate splitting in the energy levels of the system. By working in a regime where the valleys are far apart and the particle's journey is slow, the authors showed that the complex, wave-like "jumps" predicted by the first method are exactly the same as the statistical sum of the particle's journeys in the second method. They demonstrated that the mysterious mathematical shifts that occur when the particle's path crosses a critical threshold are not just abstract artifacts, but correspond directly to the physical act of the particle turning around in a valley and retracing its steps.

The researchers began by constructing a model of a one-dimensional world containing a series of identical valleys separated by hills. In this setting, a particle can tunnel from one valley to the next, and if it has enough energy, it can visit many valleys in a single cycle. The team analyzed the "instanton," a specific type of tunneling event where the particle crosses a single hill. When multiple such events happen in sequence, they form a chain of visits. The authors calculated the total contribution of these chains to the system's energy by integrating over the time intervals between each crossing. This calculation, known as a quasi-zero mode integral, allowed them to treat the time gaps between tunneling events as flexible variables rather than fixed points.

A key discovery was how the mathematics of these time intervals relates to the wave-like description. The researchers found that the time intervals between tunneling events could be mapped onto specific paths in a complex mathematical space. When the particle's journey involves a "turning back" event—where it crosses a hill, enters a valley, and immediately crosses back—the mathematical path for that interval has a special property. It can be viewed as a combination of two distinct routes that differ by a single loop around a critical point. This loop, which the authors call a Hankel cycle, is the geometric source of the "Stokes jump," a sudden change in the mathematical description that occurs when the system crosses a specific boundary. By showing that the difference between the two possible descriptions of the particle's state is exactly equal to this single loop, the team proved that the wave-like jumps are a direct consequence of the particle's ability to turn around in a valley.

The study also revealed a surprising limitation in how much information is preserved when we look at the final energy of the system. When the particle performs a long, complex journey visiting many valleys in a specific order, the final energy depends only on the total number of times it crossed each hill and the total number of times it turned around. The specific order in which these events happened is lost. The researchers constructed two different sequences of events that visit the valleys in a different chronological order but result in the exact same energy contribution. This means that for a system with three or more valleys, the final energy spectrum cannot tell us the exact history of the particle's journey, only the aggregate statistics of its crossings and turns. This loss of ordering information is a fundamental feature of the system at this level of approximation.

To confirm their theoretical findings, the authors applied their method to a specific case of a triple-well potential, a landscape with three identical valleys. They calculated the predicted energy splitting between the lowest energy states using their new method and compared it with a direct numerical solution of the Schrödinger equation, the fundamental equation governing quantum motion. The results matched with high precision as the quantum effects became smaller, validating their approach. This agreement confirms that the complex interplay between the wave-like description and the particle-like description is consistent and that the mathematical tools used to describe tunneling in multi-valley systems are robust.

The work provides a clear, unified picture of how quantum tunneling works in complex landscapes. It shows that the seemingly abstract jumps in the wave description are rooted in the physical geometry of the particle's path. It also clarifies that while the detailed history of a particle's journey is lost in the final energy measurement, the essential counts of its crossings and turns are preserved. This understanding helps physicists reconcile different mathematical frameworks and offers a more complete view of how quantum particles navigate the intricate terrain of the microscopic world.

Technical Summary: Multi-Instantons and Exact-WKB Stokes Jumps in Multi-Well Quantum Mechanics

Problem Statement
This paper investigates the precise correspondence between two complementary descriptions of tunneling in one-dimensional quantum mechanics with $N$ equal-depth minima: the exact-WKB (Wentzel–Kramers–Brillouin) formalism and the multi-instanton expansion of the Euclidean path integral. While the relationship between instanton–anti-instanton ambiguities and Borel ambiguities in perturbation theory is well-established for single barriers (the Bogomolny–Zinn-Justin or BZJ cancellation), the authors address three specific questions regarding multi-well systems:

Can the exact-WKB prediction for the connected spectral coefficient of an arbitrary consecutive barrier interval be reproduced by the quasi-zero mode integral (QZI) of the path integral, including the specific lateral (Stokes) jump?
Does the QZI contour relation (Picard–Lefschetz transformation) at the instanton–anti-instanton ( $I\bar{I}$ ) pairs reproduce the Delabaere–Dillinger–Pham (DDP) Stokes automorphism acting on the exact-WKB coefficients?
At higher nonperturbative grades, does the scalar spectral data retain enough information to reconstruct the full ordered sequence of tunneling events, or is there a loss of ordering information (degeneracy) when projecting to scalar coefficients?

Methodology and Regime
The analysis is conducted within a specific "fixed-grade large- $T$ " regime. The authors fix a finite nonperturbative grade $K$ (the number of barrier weights in a term) and consider the Euclidean time $T$ to be parametrically large, satisfying $K \log(1/|\hbar|) \lesssim T \ll \Delta_{np}^{-1}$ , where $\Delta_{np}$ is the exponentially small level splitting. This window ensures that instanton events are widely separated (dilute gas approximation) while avoiding the need to resum the entire transseries.

The methodology proceeds in two parallel tracks that are later matched:

Exact-WKB Track: The authors utilize established results for the lateral quantization conditions $Q^\pm_N$ and the DDP Stokes automorphism. They derive the connected coefficient $C^\pm_I$ for a consecutive barrier interval $I=[a,b]$ by expanding the reduced quantization function in barrier weights. This coefficient depends on local perturbative oscillations $u^\pm_v$ at the outer wells and a lateral-invariant internal factor $K_I$ .
Path Integral Track: The authors construct the QZI directly from a regular dilute neighborhood of the multi-instanton configuration in the complexified path space. They integrate out normal fluctuations (one-loop) and reduce the integral over collective coordinates (separations $\tau_r$ ) to Gamma and Hankel periods. Crucially, they identify the relevant energy scale with the "Weber regime" (parabolic cylinder equation) near the double turning points, rather than the generic Airy regime.

Key Contributions and Results

Reproduction of Connected Coefficients and Stokes Jumps:
The authors demonstrate that the QZI, when normalized by local Weber connection matrices, reproduces the exact-WKB connected coefficients $C^\pm_I$ exactly.
- Normalization: The normalization of the repulsive separations (internal wells) is fixed by matching to the off-diagonal entry of the local Weber connection matrix. This fixes the normalization for attractive separations (outer wells) without additional input.
- Contour Relation: The two lateral integration contours for an attractive separation ( $I\bar{I}$ pair) differ by exactly one Hankel cycle ( $H$ ). Specifically, the lower contour $\tilde{c}_-$ is related to the upper contour $c_+$ by the linear Picard–Lefschetz relation $\tilde{c}_- = c_+ - H$ .
- Stokes Jump: Applying the normalized period map to this linear relation yields the complementary rule $u^-_v = 1 - u^+_v$ . For a two-barrier interval, the product of these local relations at the two outer wells reproduces the four-term DDP jump of the connected coefficient: $C^-_I = C^+_I - K_I(u^+_a + u^+_c - 1)$ . This confirms that the DDP Stokes jump is the period image of the linear Picard–Lefschetz transformation of the QZI cycles.
Ordering Degeneracy at Higher Grades:
The paper establishes that the map from ordered event sequences to normalized scalar spectral coefficients is not one-to-one at higher fixed grades.
- The scalar weight of a closed event sequence depends only on the aggregate counts $(k, R, X)$ of barrier crossings, $I\bar{I}$ pairs (turning points), and $II/\bar{I}\bar{I}$ pairs (pass-throughs), rather than the chronological order of events.
- The authors construct explicit examples of inequivalent event sequences (e.g., at grade six in a triple-well potential) that possess identical counts and thus produce identical scalar weights. This "ordering degeneracy" implies that the scalar spectrum loses information about the specific ordering of tunneling events.
- This degeneracy is shown to be a property of the leading dilute, one-loop approximation and is inherited by the spectral coefficient. It persists for any potential with $N \ge 3$ wells.
Transfer Matrix Formulation:
The authors construct a transfer matrix acting on oriented tunneling events using the local Weber amplitudes. The determinant of this matrix reproduces the exact-WKB function $Z^\pm_N$ , and its trace expansion generates the scalar weights of all closed event sequences. This formalism explicitly demonstrates how the inclusion-exclusion principle in the exact-WKB expansion corresponds to the sum over different event orderings in the path integral.

Significance and Claims
The paper claims to provide a rigorous, coefficient-by-coefficient verification of the correspondence between exact-WKB Stokes phenomena and multi-instanton path integrals in multi-well systems, extending the known single-barrier BZJ cancellation to arbitrary consecutive barrier intervals.

Unification of Descriptions: It shows that the nonlinear DDP transformation of exact-WKB symbols arises from linear Picard–Lefschetz transformations of integration cycles in the path integral, provided one works within the local Weber regime and fixes normalizations via the local connection matrix.
Information Loss: It identifies a fundamental limitation in reconstructing the full nonperturbative structure from scalar spectral data alone: the ordering of tunneling events is lost at higher grades due to the degeneracy of scalar weights for distinct event sequences.
Scope and Limitations: The authors are explicit that their results hold within the regular dilute, one-loop, local Weber approximation. They do not claim to derive the full DDP action (specifically the nonlinear factors $(1+B_e)^{\pm 1}$ ) solely from the QZI; rather, they show the compatibility of the reduced contour relations with the DDP jump, assuming the exact-WKB input for the perturbative factors. They also note that the absolute normalization of elementary blocks is fixed by the local Weber equation rather than derived from the path integral for general potentials (though checked for the symmetric double well).

The work is presented as a controlled laboratory for the general problem of resurgence, using the simplicity of quantum mechanics to clarify the geometric and algebraic structures (Picard–Lefschetz theory, cluster algebras) underlying the cancellation of ambiguities in nonperturbative physics.

Technical Summary: Multi-Instantons and Exact-WKB Stokes Jumps in Multi-Well Quantum Mechanics

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