Subgap in the surface bound states spectrum of Superfluid $^3$He-B with Rough Surface

Subgap structure in the surface bound states spectrum of Superfluid $^3$He-B with rough surface is discussed. The subgap is formed by the level repulsion between the surface bound state and the continuum states in the course of multiple scattering by the surface roughness. We show that the level repulsion is originated from the nature of the wave function of the surface bound state that is now recognized as Majorana Fermion. We study the superfluid $^3$He-B with a rough surface and under magnetic field perpendicular to the surface using the quasi-classical Green function together with random S-matrix model. We calculate the self-consistent order parameters, the spin polarization density and the surface density of states. It is shown that the subgap is found also under the magnetic field perpendicular to the surface. Magnetic field dependence of the transverse acoustic impedance is also discussed.