Nonperturbative Formulation of Resonances in Quantum Mechanics Based on Exact WKB Method

Abstract

We study quasi-stationary states in quantum mechanics using the exact Wentzel--Kramers--Brillouin (WKB) analysis as a nonperturbative framework. Whereas previous works focused 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 nonpolynomial potentials.