Quantum Reflection Amplitude for the Double Sine-Gordon Kink-Antikink System: From the geometric subsystem programme to the Born approximation

We apply the geometric subsystem quantisation programme to the double sine–Gordon (DSG) model and derive the first Born approximation for the quantum reflection amplitude of a kink–antikink pair in the asymptotic regime. The exact canonical symplectic form and Hamiltonian on the two‑body collective‑coordinate ansatz are taken from the companion paper~{CCsymplectic}. The perturbative Darboux transformation constructed in~{Darboux} is used to diagonalise the off‑diagonal symplectic couplings. We prove that, to leading order in the small parameter ε = e-μ s (s is the kink–antikink separation), the Darboux correction does not alter the static interaction potential. In the non‑relativistic limit, the relative motion reduces to a one‑dimensional Schr\"odinger equation with the exact static potential Vᵢₙₜ(r). The first Born approximation yields the reflection amplitude expressed as an integral involving this potential. The low‑energy limit of the amplitude is determined by the total integrated strength of the interaction. The derivation provides the first Born approximation for the quantum reflection amplitude of DSG kink–antikink scattering within the geometric subsystem programme.