Existence of Majorana bound states in a superconducting nanowire near an impurity

We consider a nanowire with the $s$-wave superconducting order induced as a result of the proximity effect in the presence of the Zeeman field and the Rashba interaction. For a small superconducting gap and small momenta, we analytically prove the existence of Majorana bound states for a certain local change in the Zeeman field or the superconducting order and also obtain explicit expressions for the corresponding wave functions. We study the scattering of excited states with energies that are close to boundary gap points in the case of propagation through an impurity for local changes in the indicated system parameters near this impurity and show that the transmission probability is equal to unity.