Electron scattering at the domain wall

We study the transmission and reflection coefficients for the Hamiltonian of a spin-polarized electron passing through the domain wall in a ferromagnetic quantum wire. We prove that total reflection occurs for energies $\lambda\in (-\alpha,\alpha)$ ($-\alpha$ is the boundary of the essential spectrum) for both sufficiently small and sufficiently large $\lambda$, which agrees with the ballistic magnetoresistance effect in ferromagnetic nanocontacts. For energies $\lambda>\alpha$, almost total reflection becomes almost total transmission, and both effects occur without a spin flip.