On continuum and resonant spectra from exact WKB analysis

• URL: http://arxiv.org/abs/2508.09211v2
• Date: Sat, 11 Oct 2025 08:15:20 GMT
• Title: On continuum and resonant spectra from exact WKB analysis
• Authors: Okuto Morikawa, Shoya Ogawa,
• Abstract summary: We employ the complex scaling method (CSM) in conjunction with exact WKB analysis to elucidate the geometric structure of scattering problems.<n>We derive the S-matrix for the inverted Rosen--Morse potential and reveal its underlying complex-geometric features.<n>Our analysis bridges scattering cross-sections and spectral theory, offering new geometric insights into quantum resonance and scattering phenomena.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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