Numerical Framework for the Quantum S‑Matrix of the Double Sine-Gordon Kink-Antikink System: Semi‑analytic method and physical expectations

We present a detailed numerical framework for computing the full quantum S‑matrix of the kink–antikink pair in the double sine–Gordon model, building on the non‑perturbative scattering formalism. The exact static interaction potential, obtained from the geometric subsystem programme, is evaluated numerically from the static kink profile. The half‑line Schrodinger equation with Neumann boundary condition is integrated outward and matched to the exactly solvable exponential tail using Bessel functions of purely imaginary order. We describe the shooting method for the static kink, the numerical quadrature for the potential, the ODE integration algorithm, and the matching procedure. The method is complete and ready for implementation; the expected qualitative behaviour of the reflection amplitude, scattering phase shift, and low‑energy parameters is discussed based on known properties of the potential. Numerical results will be reported in a separate publication.