双阱势的复现：双重井
The Double Well Done Doubly-Well
AI总结：
本文通过精确WKB和欧几里得路径积分两种互补方法，以统一符号和显式计算至四瞬子级和三圈阶，给出了对称双阱势能级瞬子展开的完整自洽描述，揭示了微扰与非微扰物理的深层纠缠。
摘要：
对称双阱势是最简单的量子力学系统之一，其中微扰和非微扰物理深度纠缠。其能级具有阱间分离负幂次展开的非解析形式，系数呈阶乘增长，而宇称分裂是指数小的，对微扰论不可见。复现将这两个特征联系在一起，将精确谱组织成一个严格约束的瞬子展开。本文从两种互补方法——精确WKB和欧几里得路径积分——给出了该瞬子展开的自洽描述，采用统一符号并进行了至四瞬子级和三圈阶的显式计算。在精确WKB中，由Delabaere-Dillinger-Pham关系编码的Stokes现象控制了波函数经过转折点的解析延拓。用Voros符号表示的量子化条件决定了完整的瞬子展开。DDP关系是局域的，不需要知道能量面的全局拓扑，但该能量面是椭圆曲线。在路径积分中，椭圆曲线以不同方式出现：经典鞍点是欧几里得时间的双周期椭圆函数，而Stokes现象在准零模的有限维流形中发生，而非通过波函数的解析延拓。Lefschetz thimble分解决定了哪些鞍点有贡献，得到的配分函数瞬子展开比能量瞬子展开简单得多：在每个瞬子阶，T依赖性由准零模thimble积分固定的多项式给出。两种方法以互补方式部署共享的数学基础设施，表明双阱势是探索复现的理想环境。
英文摘要：
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.
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