Tunneling in the presence of phonons: A solvable model

We obtain the solution for an electron tunneling through a thin potential barrier with local Einstein phonons, by means of a continued fraction expansion. Our results demonstrate the strong feedback between inelastic and elastic scattering, the associated singularities, an increase in low-energy transmission with a nonmonotonic temperature dependence, and a crossover from quantum to adiabatic behavior with increasing temperature and electron velocity.