Exact WKB analysis of inverted triple-well: resonance, PT-symmetry breaking, and resurgence

TL;DR

This paper applies exact WKB analysis to an inverted triple-well potential in non-Hermitian quantum mechanics, revealing resonance phenomena, PT-symmetry breaking, and the role of resurgence in spectral properties.

Contribution

It derives exact quantization conditions, constructs trans-series solutions, and links resurgence structures to physical phenomena like PT-symmetry breaking and exceptional points.

Findings

Identifies explicit conditions for PT-symmetry breaking.

Derives an algebraic relation for the exceptional point.

Shows median-summed non-perturbative corrections vanish at the exceptional point.

Abstract

We study non-Hermitian quantum mechanics of an inverted triple-well potential within the exact WKB framework. For a single classical potential, different Siegert boundary conditions define three distinct quantum problems: the PT-symmetric, resonance, and anti-resonance systems. For each case, we derive the exact quantization condition and construct the associated trans-series solution. By identifying the resurgent structures and cancellations in these non-Hermitian setups, we obtain the median-summed series, clarifying when the spectra are real or complex in accordance with the physical properties of each system. Establishing explicit links to the semi-classical path integral formalism, we elucidate the roles of bounce and bion configurations in these non-Hermitian systems. This analysis predicts PT-symmetry breaking, which we also verify numerically. Using the median quantization conditions, we prove the existence of this symmetry breaking and establish an exact equation for the exceptional point, which emerges as a remarkably simple algebraic relation between the bounce and bion actions. We further show that the median-summed non-perturbative correction to the spectrum vanishes at the exceptional point, while the resurgent structure survives through a universal minimal trans-series. For the resonance and anti-resonance systems, we find that the exact median-summed spectra are related by complex conjugation, representing time reversal in this setting, are necessarily complex, and do not exhibit an exceptional point. Although their spectra differ significantly from the PT-symmetric case, they share the same minimal trans-series. By maintaining explicit links with the path integral saddles and the formal theory of resurgence, our analysis provides a unified and general perspective on the quantization of non-Hermitian theories.