Exact WKB in all sectors II: Potentials with non-degenerate saddles

TL;DR

The paper extends Exact WKB to all spectral sectors for general one-dimensional potentials with non-degenerate saddles by analytic continuation of the energy parameter, revealing sector-specific median quantization conditions and trans-series structures. It provides a Weber-type dictionary and Stokes-analysis framework to connect perturbative and non-perturbative sectors, and applies it to asymmetric triple-well and tilted double-well systems, uncovering bion/complex saddles, cluster expansions, and spectrum-realization mechanisms via median summation. It also develops transformation rules for P-NP relations in genus-1 systems, clarifying how classical parameters such as frequencies and instanton actions govern duality-induced changes across saddles. Collectively, the results strengthen the link between EWKB and path-integral resurgence, and demonstrate predictive power for non-perturbative spectra, including PT-symmetric cases and potential higher-genus extensions. These insights advance the systematic understanding of non-perturbative quantum mechanics through exact quantization and resurgence.

Abstract

We discuss the exact quantization of general one-dimensional potentials in view of the exact-WKB formalism. Building on our previous work, we perform analytic continuations across different sectors via the complexification to the spectral (energy) parameter $u$ and identify continuous and discontinuous transitions of the exact spectrum for generic potentials. When the transition is discontinuous, it is characterized by the Stokes phenomena, inducing different exact (median) quantization conditions, thereby distinct trans-series structures valid in different sectors. We analyze two illustrative examples, namely asymmetric triple-well (ATW) and tilted double-well (TDW), and verify the general qualitative analysis by deriving exact (median) quantization conditions in each sector. Moreover, by obtaining the trans-series solutions for each system, we identify bion/bounce configurations and show that the trans-series of ATW is organized in accordance with the cluster expansion of the bion gas and there should exist a previously neglected complex saddle in the TDW system. These identifications further strengthen the link between path integral and exact-WKB formalisms, while also demonstrating the predictive power of the latter. In parallel, for the P-NP relations of genus-1 systems, we derive transformation rules between any perturbative and non-perturbative pair of WKB-cycles. Our results show that the entire resurgence data of a genus-1 system transforms only by the change of classical parameters, i.e. frequencies and bion/bounce actions, and the perturbative energy series. This also reveals the underlying reasons of the previously found $S$-duality transformations.

Figures (18)

• Figure 1: Weber type perturbative and non-perturbative cycles for all the possible cases we encounter in one dimensional problems.
• Figure 2: The change of the Stokes diagrams during the transition across a barrier top which is induced by $\theta_u: 0 \rightarrow \mp\pi$. It demonstrates the continuous character of the transition as no Stokes phenomenon is encountered. As a result, the transition matrices and the quantization conditions stay intact. (This figure first appeared in Misumi:2024gtf.)
• Figure 3: The rotation of Stokes line emerging from the turning points at $x=x_\pm$. Note that these figures first appeared in Misumi:2024gtf. Here, only the labels for $x=x_\pm$ is altered.
• Figure 4: Change of the Stokes geometry during the transition below a locally harmonic well with a neighbouring barrier. Contrary to the transition across a barrier top, we encounter two Stokes automorphisms at $\theta_2=\theta_1$ and $\theta_u =\theta_2$, which leads to distinct exact quantization conditions, namely trans-series solutions at $\theta_u=0$ and $\theta=\mp\pi$.
• Figure 5: An illustration of a locally harmonic potential with a spectrum contains disconnected sectors associated with different trans-series representations. Each distinct sector is associated with a different trans-series and is represented by a different color, which are separated by gray lines. Critical values of the discontinuous transitions are labeled with $u_1$, $u_2$ and $u_4$. At other critical values $u_3$, $u_5$ and $u_6$, the transition is continuous as they correspond to barrier tops.