Exact WKB as unified analytic structure for resonance physics

We develop a unified framework for analyzing quantum mechanical resonances using the exact WKB method. The non-perturbative formulation based on the exact WKB method works for incorporating well-established phenomenological regularizations, the ABC theorem (proof of the completeness of Hilbert space), and the rigged Hilbert space in resonant phenomena. By examining the inverted Rosen-Morse potential, we illustrate how the exact WKB analysis captures resonant phenomena rigorously. Also, we clarify the corresponding linear spaces defined in each step of the exact WKB manipulations. The complementarity between the essential analyticity for resonance and the ABC theorem leads us to construct a modified Hilbert space called the rigged Hilbert space within the exact WKB framework. This offers a deeper understanding of resonant states and their analytic structures. Our results provide a concrete demonstration of the non-perturbative accuracy of exact WKB methods in unstable quantum systems.