Inelastic quantum tunneling through disordered potential barrier

The effect of inelastic scattering on quantum tunneling through a rectangular potential barrier, of length $L$, containing randomly distributed impurities, is considered. It is shown that, despite the fact that the inelastic transition probability $\mathcal{T}_{\mathrm{inelastic}}$ is exponentially small, it may be greater than that of the elastic in the out of resonance conditions for some energy interval $\triangle E$ (the energy of the incident particle is far from the spectrum of the random Hamiltonian). The asymptotic of the tunnelling transition probability for various impurity configurations are found and the energy interval $\triangle E$ where $\mathcal{T}_{\mathrm{elastic}}\sim\exp(-L/L^{*})\lesssim\mathcal{T}_{\mathrm{i nelastic}}$ is determined.