Cheshire Cat复苏、自复苏与拟精确可解系统
Cheshire Cat resurgence, Self-resurgence and Quasi-Exact Solvable Systems
摘要：
本文研究了量子力学Sine-Gordon势和双阱势的单参数$ζ$ -形变，我们分别称之为双Sine-Gordon势（DSG）和倾斜双阱势（TDW）。在这些系统中，对于$ζ$的正整数值，最低的$ζ$状态被证明是精确可解的DSG -一个被称为准精确可解性（QES）的特征-和可解的TDW在微扰理论的所有订单。对于DSG这样的状态不显示任何瞬变的依赖于耦合常数，虽然行动有真正的鞍。另一方面，虽然它没有真正的鞍点，TDW承认所有的订单微扰状态是不可正规化的，因此，需要一个非微扰的能量转移。这两个难题都可以通过包含复杂的鞍来解决。我们表明，收敛性是由隐藏的拓扑角的量化。此外，我们认为，QES系统可以联系到精确取消的真实和复杂的非微扰鞍的所有订单的半经典扩展。我们还表明，整个复苏的结构仍然编码在解析性质的$ζ$ -变形，即使正好在整数值的$ζ$复苏的机制是模糊的缺乏歧义的波莱尔和扰动理论以及非微扰的贡献。通过这种方式，所有的复苏的特点仍然存在，即使它的作用似乎消失了，很像挥之不去的笑容的柴郡猫。我们还表明，微扰级数是自我复苏的-一个功能，有一个一对一的关系之间的早期条款的微扰膨胀和后期条款的同一膨胀-这是密切相关的邓恩-Ünsal关系。我们明确地证实，情况确实如此。
英文摘要：
We explore a one parameter $ζ$-deformation of the quantum-mechanical Sine-Gordon and Double-Well potentials which we call the Double Sine-Gordon (DSG) and the Tilted Double Well (TDW), respectively. In these systems, for positive integer values of $ζ$, the lowest $ζ$ states turn out to be exactly solvable for DSG - a feature known as Quasi-Exact-Solvability (QES) - and solvable to all orders in perturbation theory for TDW. For DSG such states do not show any instanton-like dependence on the coupling constant, although the action has real saddles. On the other hand, although it has no real saddles, the TDW admits all-orders perturbative states that are not normalizable, and hence, requires a non-perturbative energy shift. Both of these puzzles are solved by including complex saddles. We show that the convergence is dictated by the quantization of the hidden topological angle. Further, we argue that the QES systems can be linked to the exact cancellation of real and complex non-perturbative saddles to all orders in the semi-classical expansion. We also show that the entire resurgence structure remains encoded in the analytic properties of the $ζ$-deformation, even though exactly at integer values of $ζ$ the mechanism of resurgence is obscured by the lack of ambiguity in both the Borel sum of the perturbation theory as well as the non-perturbative contributions. In this way, all of the characteristics of resurgence remains even when its role seems to vanish, much like the lingering grin of the Cheshire Cat. We also show that the perturbative series is Self-resurgent -a feature by which there is a one-to-one relation between the early terms of the perturbative expansion and the late terms of the same expansion -which is intimately connected with the Dunne-Ünsal relation. We explicitly verify that this is indeed the case.
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