Reduced Quantum Scattering of the Double Sine-Gordon Kink-Antikink System - Non‑perturbative treatment within the collective‑coordinate effective mode

We formulate the non‑perturbative quantum scattering problem for the kink–antikink pair in the double sine–Gordon model using the exact static interaction potential derived from the geometric subsystem quantisation programme. The potential is obtained as an overlap integral of the static kink profile and is smooth, attractive, and exponentially decaying. The relative motion is governed by a one‑dimensional Schrodinger equation on the half‑line. We express the reflection amplitude in terms of the Jost function and set up a semi‑analytic matching method that exploits the exactly solvable exponential tail of the potential. The formulation is fully rigorous and provides a complete theoretical framework for numerical computation of the full S‑matrix. A brief discussion of bound states and the low‑energy effective range expansion is included.