On continuum and resonant spectra from exact WKB analysis

Resonance phenomena are central to many quantum systems, where resonant states are typically described by pole singularities of S-matrix. In this work, we apply the complex scaling method (CSM) and exact WKB analysis to describe scattering problems, incorporating both bound and resonant states. We compute continuum spectrum based on the exact WKB analysis and derive the S-matrix for the inverted Rosen--Morse potential. We reinterpret the Aguilar--Balslev--Combes theorem, on which CSM is based; we then discuss the physical significance of the Siegert boundary condition and rigorously define physical states in a modified Hilbert space. Our analysis connects the scattering cross-section and spectral theory, providing insights into the scattering theory and related formulas.