所有扇区中的精确 WKB 方法 I：具有简并鞍点的势
Exact WKB in all sectors I: Potentials with degenerate saddles
AI总结：
本文提出一种对能量参数 $u$ 进行复化的新精确 WKB 方法，解决了多扇区局部谐和势的精确量子化问题，揭示了量子作用量中的 resurgence 结构与 $S$-对偶性。
摘要：
我们通过分析 Airy 和 Weber 类型探索了精确 WKB（EWKB）方法，重点关注多扇区中局部谐和势的精确量子化。我们工作的核心创新在于对能量参数 $u$ 引入了一种新颖的复化方法，这与 Borel 可和性中常用的（半经典）展开参数的复化不同。这种新技术允许在势的不同扇区之间进行连续的解析延拓，同时保持精确量子化条件，甚至在中位求和之前也是如此。通过在势垒顶部上方重新定义 $A$-循环，我们确保量子化条件保持为实数，并通过使用 Stokes 自同构和中位重求和，证明了 resurgence 结构在扇区间的转换中得以保持。此外，我们讨论了 Weber 型精确 WKB 方法，为所有类型鞍点附近的量子作用量提供了精确估计，推广了以往的结果。通过分析这些量子作用量，我们揭示了 $S$-对偶性的存在，促进了微扰与非微扰行为之间的交换，并猜想对偶理论间 P-NP 关系的映射。我们的研究涵盖了周期势和对称双井势，证明了精确 WKB 方法能够捕捉所有扇区量子系统中复杂的结构，包括多瞬子贡献和量子作用量的 resurgence。
英文摘要：
We explore the exact-WKB (EWKB) method through the analysis of Airy and Weber types, with an emphasis on the exact quantization of locally harmonic potentials in multiple sectors. The core innovation of our work lies in introducing a novel complexification approach to the energy parameter $u$, distinct from the common complexification of the (semi-classical) expansion parameter used in Borel summability. This new technique allows for continuous analytical continuation across different sectors of a potential while maintaining the exact quantization condition, even before median summation. By redefining the $A$-cycle above the potential barrier top, we ensure the quantization condition remains real and, by use of the Stokes automorphism and the median resummation, show that the resurgence structure is preserved across transitions between sectors. Furthermore, we discuss the Weber-type exact-WKB method, offering exact estimates for quantum actions around all types of saddle points, generalizing previous results. Through the analysis of these quantum actions, we reveal the presence of $S$-duality, facilitating the exchange between perturbative and non-perturbative behaviors, and we conjecture the mapping of the P-NP relations between dual theories. Our study encompasses periodic and symmetric double-well potentials, demonstrating that the exact-WKB method captures intricate structures in quantum systems in all sectors, including multi-instanton contributions and the resurgence of quantum actions.
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