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arXiv:2608.23342

Instantons in a Double-Well are Poisson Distributed

Jacob Shapiro

math-phmath.APmath.PRquant-phSubmitted Aug 24, 2026• Announced Aug 25, 2026
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Abstract

We give a rigorous realization of the dilute instanton picture for a semiclassical Schrödinger operator with a symmetric double-well potential on $\mathbb{R}^n$. Using a localized Feynman--Kac representation, we decompose the heat-kernel trace according to the number of passages made by a Brownian bridge between shrinking neighborhoods of the two wells. We identify the weight of one passage with a hopping coefficient $ρ_λ$, $\displaystyle ρ_λ= \int_{\partialΩ} \left( \nabla\overline{\varphi_{λ,0}^Ω}\, \varphi_{λ,0}^{-Ω} - \overline{\varphi_{λ,0}^Ω}\, \nabla\varphi_{λ,0}^{-Ω} \right)\cdotν. $ On the exponentially long time scale $β=N/\lvertρ_λ\rvert$, the number of passages converges, for every fixed $N>0$, to a Poisson random variable of mean $N$. We identify $-\frac{1}λ\log\lvertρ_λ\rvert\to S(d,-d)$ and obtain $\displaystyle E_1(λ)-E_0(λ) = 2\lvertρ_λ\rvert\left(1+o(1)\right). $ Thus the familiar instanton expansion of the double-well eigenvalue splitting emerges directly from a factorization of the heat-kernel trace.

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We give a rigorous realization of the dilute instanton picture for a semiclassical Schrödinger operator with a symmetric double-well potential on Rn\mathbb{R}^nRn. Using a localized Feynman--Kac representation, we decompose the heat-kernel trace according to the number of passages made by a Brownian bridge between shrinking neighborhoods of the two wells. We identify the weight of one passage with a hopping coefficient ρλρ_λρλ​,
ρλ=∫∂Ω(∇φλ,0Ω‾ φλ,0−Ω−φλ,0Ω‾ ∇φλ,0−Ω)⋅ν.\displaystyle ρ_λ= \int_{\partialΩ} \left( \nabla\overline{\varphi_{λ,0}^Ω}\, \varphi_{λ,0}^{-Ω} - \overline{\varphi_{λ,0}^Ω}\, \nabla\varphi_{λ,0}^{-Ω} \right)\cdotν. ρλ​=∫∂Ω​(∇φλ,0Ω​​φλ,0−Ω​−φλ,0Ω​​∇φλ,0−Ω​)⋅ν.
On the exponentially long time scale β=N/∣ρλ∣β=N/\lvertρ_λ\rvertβ=N/∣ρλ​∣, the number of passages converges, for every fixed N>0N>0N>0, to a Poisson random variable of mean NNN. We identify −1λlog⁡∣ρλ∣→S(d,−d)-\frac{1}λ\log\lvertρ_λ\rvert\to S(d,-d)−λ1​log∣ρλ​∣→S(d,−d) and obtain
E1(λ)−E0(λ)=2∣ρλ∣(1+o(1)).\displaystyle E_1(λ)-E_0(λ) = 2\lvertρ_λ\rvert\left(1+o(1)\right). E1​(λ)−E0​(λ)=2∣ρλ​∣(1+o(1)).
Thus the familiar instanton expansion of the double-well eigenvalue splitting emerges directly from a factorization of the heat-kernel trace.
Relaxation to equilibrium of a drifted Brownian motion is quantified by a transition probability density function, whose main (multiplicative) entry is an inferred Feynman-Kac kernel of the Schrödinger semigroup operator. Although seemingly devoid of a natural probabilistic significance (except for its explicit path integral definition), the pertinent kernel relaxes to equilibrium as well. The implicit Feynman-Kac potential V(x){\cal{V}}(x)V(x), continuous, confining and bounded from below, may take negative values. If positive, V(x){\cal{V}}(x)V(x) can be interpreted as the killing rate of the decaying diffusion process. In case of relaxing F-K kernels the killing effects are tamed (often overcompensated). The taming inavoidably appears in conjunction with the existence of the negativity subdomains of V(x){\cal{V}}(x)V(x) in RRR. If locally V(x)<0{\cal{V}}(x) < 0V(x)<0, its sign inversion −V(x)- {\cal{V}}(x)−V(x) can be interpreted as the branching (cloning, alternatively bifurcation) rate in the course of the other wise free random motion. The arising killed diffusion processes with branching, we interpret as the possible path-wise background of tamed (relaxing) Feynman-Kac diffusions. We present acomputer-assisted path-wise arguments, towards a consistency of the killing/branching taming scenario, for a number of nonlinear model systems in one space dimension. Special attention is paid to Feynman-Kac potential shapes in the double well form, where an analytic access to eigenvalues and eigenfunctions is scarce. Throughout the paper the dynamics refers to the positive real time. Since the Newton-type equations of motion for admissible classical trajectories have a Euclidean form (due to the sign inverted force term), we give a brief resume of a couple of their explicit solutions, without recourse to the Euclidean time intuitions, and the instanton lore of related quantum model systems.
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.
We study a class of PT-symmetric semiclassical Schrödinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double-well potential. In the simple well case, two of the authors have proved in \cite{BoMe14} that, when the potential is analytic, the eigenvalues stay real for a perturbation of size O(1). We show here, in the double-well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB-analysis, leading to a fairly explicit quantization condition.
We study a class of ‐symmetric semiclassical Schrödinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double‐well potential. In the simple well case, two of the authors have proved in that, when the potential is analytic, the eigenvalues stay real for a perturbation of size . We show here, in the double‐well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB‐analysis, leading to a fairly explicit quantization condition.