matrices. The time-reversal of the carrier wave functions (2) can be obtained as Spin-up electrons tunnel through the double-domain-wall structure by spin-flip-assisted transmission. When the time is reversed, the angular momentum of the electrons, here the spin, is reversed. As time flows backwards, spin-down electrons tunnel backward through the double-domain-wall structure by spin-flip-assisted transmission. The time-reversed scatter- ing matrix is the Hermitian conjugate of the original one with the transmission probability exactly identical. The nature of DWs is that of an angular momentum. Time reversal re- verses the sign of the localized moments, forming thus an anti-double-domain-wall structure with a spin-down well between the two domain walls. Outside the domain wall structure, the time-reversed magnetic nanowire favors then a transport of spin-down electrons. In the adiabatic limit, the applied external magnetic field is effectively incorporated in the behavior of the DWs. Or it is equivalent to say that the effect of the time-reversal transformation reverses the direction of the magnetic field and that reverses the magnetic configuration. To contrast this situation with what is established in the literature we recall that in Ref.[2] Brouwer considered a chaotic quantum dot with the conclusion that the distribution of the pumped current for systems with TRS and those without TRS are remarkably different. In our consideration the strength of the particle flow is conserved through tunneling, which is required by the continuity equation for the Schr¨ odinger equation. The time reversed tun- neling would generate exactly the same pumped current with the electron spin reversed. In our case, the distribution of the pumped current is not a defined quantity or is a constant. Under the assumption that the scattering properties follow the time-dependent potentials instantaneously, calculating the parametric derivative of the scattering matrix is a numer- ical issue as demonstrated in the literature 3,4 including Brouwer’s seminal approach $^{2}$. In our work, the stationary transmission and reflection coefficients were analytical obtained along with their parametric derivatives. The derivative of the scattering matrix can be in- ferred from the stationary Hamiltonian: an infinitesimal change in the external parameters (here, the magnetization field and the Fermi energy) alters the Hamiltonian and hence the scattering matrix. Accordingly, in the adiabatic limit, we can find the derivative without in- 7 ψ$_{k}$ ( z ) = [( e − ikz + R $^{∗}$e ikz ) |↓〉 + R ∗ $_{f}$e κz |↑〉 ] e $^{iEt}$, z < − L, ψ$_{k}$ ( z ) = [ ( A $^{∗}$e κz + B $^{∗}$e− $^{κz}$) |↓〉 + ( C $^{∗}$e− ikz + D $^{∗}$e ikz ) |↑〉 ] e $^{iEt}$, | z | < L, ψ$_{k}$ ( z ) = T $^{∗}$e− ikz |↓〉 + T ∗ $_{f}$e − κz |↑〉 ] e $^{iEt}$, z > L. (8)