magnetization field M ( z ) with a strength determined by the Kondo-type coupling constant J . The single-particle Hamiltonian reads then 26 where Θ = JM and n$_{x}$$_{(}$$_{z}$$_{)}$ is the x ( z ) component of n ( z ), and m is the carrier’s effective mass. Θ is tunable externally, e.g. by a magnetic field. The potential profile is shown in Fig.1. The carriers’ wave functions are expressible as where |↑〉 ( |↓〉 ) is the spin-up (spin-down) component of the carrier states, k = [2 m ( E + JM )] 1 / $^{2}$/ ℏ , and κ = [2 m ( JM − E )] 1 / $^{2}$/ ℏ . As illustrated in Fig.1, we measure the electron energy E from the midpoint between spin-up and spin-down band edges. The chemical potential µ is set by the particle density and can be tuned by an external gate X$_{1}$ . The non-spin-flip (spin-flip) transmission and reflection coefficients T and R ( T$_{f}$ and R$_{f}$ ) as well as the constants A , B , C , and D can be deduced from the solutions of Eq. (1) and from the wave function continuity requirements. We note, that instead of using the wave function we can equivalently utilize the transmission and reflection amplitudes $^{29}$. For the system depicted in Fig.1 the transmission and the reflection amplitudes T (Θ , µ ) and R (Θ , µ ) were derived and given explicitly in Ref. 24 (same applies to ( T$_{f}$ and R$_{f}$ )). Following the standard scattering approach 2,3,30 we introduce the fermionic creation and annihilation operators for the carrier scattering states. The operator ˆ a † $_{Lσ}$( E ) or ˆ a$_{Lσ}$ ( E ) cre- ates or annihilates particles with total energy E and spin polarization σ in the left lead, which are incident upon the sample. Analogously, we define the creation ˆ b † $_{Lσ}$( E ) and annihi- lation ˆ b$_{Lσ}$ ( E ) operators for the outgoing single-particle states. For magnetic semiconductor nanowires with a moderate carrier density the chemical potential µ is tunable to be in one of the magnetically split subbands. In this case one achieves a full spin polarization of the electron gas. Therefore, the incident electrons are fully spin polarized. We note that the spin-down part of the wave function decays outside the double-domain-wall (DW) regime (cf. Fig.1). Correspondingly, only spin-up electrons tunnel through the barrier and contribute to the conductance. Furthermore, the energy is conserved during the tunneling process $^{31}$. As 4 ψ$_{k}$ ( z ) = [( e ikz + Re − ikz ) |↑〉 + R$_{f}$e κz |↓〉 ] e − iEt/ $^{ℏ}$, z < − L, ψ$_{k}$ ( z ) = [ ( Ae κz + Be − $^{κz}$) |↑〉 + ( Ce ikz + De − ikz ) |↓〉 ] e − iEt/ $^{ℏ}$, | z | < L, ψ$_{k}$ ( z ) = Te ikz |↑〉 + T$_{f}$e − κz |↓〉 ] e − iEt/ $^{ℏ}$, z > L, (2) H = − ℏ 2 2 m d 2 dz 2 − Θ n$_{z}$ ( z ) σ$_{z}$ − Θ n$_{x}$ ( z ) σ$_{x}$, (1)