On continuum and resonant spectra from exact WKB analysis

TL;DR

This paper combines exact WKB analysis with complex scaling to explore the geometric structure of quantum scattering, resonances, and spectral properties, providing new insights into the physical and mathematical foundations of resonance phenomena.

Contribution

It introduces a geometric perspective on the continuum and resonant spectra using exact WKB and complex scaling, and reinterprets key theorems in quantum scattering theory.

Findings

Derived the S-matrix for the inverted Rosen--Morse potential.

Revealed geometric structures underlying resonance phenomena.

Reinterpreted the Aguilar--Balslev--Combes theorem geometrically.

Abstract

Resonance phenomena are central to many quantum systems, where resonant states are typically characterized by pole singularities of the S-matrix. In this work, we employ the complex scaling method (CSM) in conjunction with exact WKB analysis to elucidate the geometric structure of scattering problems that encompass both bound and resonant states. By analyzing the continuum spectrum via the exact WKB framework, we derive the S-matrix for the inverted Rosen--Morse potential and reveal its underlying complex-geometric features. Furthermore, we reinterpret the Aguilar--Balslev--Combes theorem, the foundation of CSM, from a geometric perspective, and discuss the physical significance of the Siegert boundary condition within a rigorously defined modified Hilbert space. Our analysis bridges scattering cross-sections and spectral theory, offering new geometric insights into quantum resonance and scattering phenomena.