Non-perturbative formulation of resonances in quantum mechanics based on exact WKB method

Abstract: We study quasi-stationary states in quantum mechanics using the exact WKB analysis as a non-perturbative framework. While previous works focus mainly on stable systems, we explore unstable states such as resonances. As a concrete example, we analyze the inverted Rosen--Morse potential, which exhibits barrier resonance. This model allows exact solutions, enabling a direct comparison with exact WKB predictions. We provide a simple analytic picture of resonance and demonstrate consistency between exact and WKB-based results, extending the applicability of exact WKB analysis to non-polynomial potentials.