Cheshire Cat resurgence, Self-resurgence and Quasi-Exact Solvable Systems

TL;DR

This work studies a one-parameter $\zeta$-deformation of the DSG and TDW quantum-mechanical systems, revealing a rich resurgence structure that ties together Quasi-Exact Solvability, complex saddles, and hidden topological angles. By combining QES analysis, Bender-Wu perturbation theory, and complex-bion contributions, the authors demonstrate that for $\zeta\in\mathbb{N}^{+}$ the lowest states are algebraically solvable in DSG and perturbative to all orders in TDW, with nonperturbative effects accounted for by HTA-induced cancellations. They introduce Cheshire Cat resurgence to describe how resurgence persists under analytic continuation of $\zeta$, and they show self-resurgence via a Dunne-Uns al-type relation linking early and late terms of the same perturbative series. The results connect to quantum-field-theory contexts such as adjoint QCD and sigma models, suggesting implications for exact semiclassics, path-integral contours, and nonperturbative dynamics in gauge theories.

Abstract

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.

Figures (5)

• Figure 1: For integer $\zeta$, perturbation theory for lowest $\zeta$ states is convergent (e.g. $\zeta=1$ is the supersymmetric case), but others are divergent. For non-integer $\zeta$, perturbation theory for all states is asymptotic. The energy bands of the DSG system describe dependence of energy levels on the topological $\theta$ angle.
• Figure 2: An illustration of the exactly solvable states. The blue-shaded rectangles represent bands by changing the theta angle, who's width is non-perturbative and not exactly solvable for any $\zeta$. However, it is possible to solve either for the energy of the top or of the bottom of the band when $\zeta$ is an odd or an even integer respectively. Note that $\zeta=1$ case is a supersymmetric limit, and the bottom of the band corresponds to the supersymmetric ground state given by $\psi_0=e^{-W(x)}=e^{\frac{\omega}{g}\cos(x\sqrt g)}$.
• Figure 3: A plot of eigenvalues $E(\nu=\{0,1,2\},g)$ for $\zeta=3$ as a function of coupling $g$. The solid lines represent the real part of the all orders in perturbation theory result \ref{['eq:zeta3_eigenvalues']}, while the dashed lines represent the numerical solution to the Schrödinger equation. Notice that $E(\nu=1,g)$ and $E(\nu=2,g)$ in \ref{['eq:zeta3_eigenvalues']} collide and turn into complex conjugate pairs when $g=\frac{1}{3\sqrt{3}}$.
• Figure 4: A plot of perturbative eigenvalues (solid line) and the numerical solutions (dashed line) for the states $E(\nu=\{0,1,2,3\},g)$ for $\zeta=4$ as a function of coupling $g$ which are solutions of \ref{['eq:zeta4_eigenvalues']}. Notice that while $E(\nu=\{0,1\},g)$ agree quite well (up to non-perturbative corrections), the perturbative values of $E(\nu=2,g)$ and $E(\nu=3,g)$ merge at some value of $g$ and turn into complex conjugate pairs.
• Figure 5: A plot of first $\zeta$ eigenvalues to all order of perturbation theory for $\zeta=10,15,20$ (left to right).