Unified exact WKB framework for resonance -- Zel'dovich/complex-scaling regularization and rigged Hilbert space

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 the Zel'dovich regularization, the complex scaling method, and the rigged Hilbert space. While previous studies have demonstrated the exact WKB analysis in bound state problems, our work extends its application to quasi-stationary states. By examining the inverted Rosen--Morse potential, we illustrate how the exact WKB analysis captures resonant phenomena in a rigorous manner. We explore the equivalence and complementarity of different well-established regularizations \`a la Zel'dovich and complex scaling within this framework. Also, we find the most essential regulator of functional analyticity and construct a modified Hilbert space of the exact WKB framework for resonance, which is called the rigged Hilbert space. 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.