Scattering Problem in Bose-Einstein Condensates with Magnetic Domain Wall

We present a comprehensive theoretical study of linear wave scattering from magnetic domain walls with varied twist angles $\Theta$ in spin-$1/2$ Bose-Einstein condensates (BECs). Using a gauge transformation, we show that scattering observables depend solely on the total twist $\Theta$, independent of chirality. Within the Bogoliubov-de Gennes (BdG) framework, we develop a transfer-matrix method to compute reflection and transmission coefficients for incident phonons and free particles. Our results reveal a scattering threshold at the Zeeman energy $E = \hbar\Omega_0$, separating a pure phonon regime from multi-channel scattering involving both collective and single-particle excitations above threshold. For large twist angles, competition between kinetic and Zeeman energies reduces the effective spin rotation, leading to comb-like density modulations and Fano-like resonances below threshold. The transition probability between phonon and particle channels is strongly tunable with $\Theta$, enhanced for odd multiples of $\pi$ but suppressed for even multiples. These findings establish twist-engineered domain walls as a versatile platform for controlling quantum transport, with implications for atomtronic devices and quantum simulation.