  We study an electron that interacts with phonons or other linear or nonlinear
excitations as it resonantly tunnels. The method we use is based on mapping a
many-body problem in a large variational space exactly onto a one-body problem.
The method is conceptually simpler than previous Green's function approaches,
and allows the essentially exact numerical solution of much more general
problems. We solve tunneling problems with transverse channels, multiple sites
coupled to phonons, and multiple phonon degrees of freedom and excitations.
