Electronic transport through a local state linearly coupled to phonon modes

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A resolvent formalism is applied to the problem of inelastic scattering of an electron linearly coupled to a set of phonon modes. It is shown how the many phonon mode coupling and excitation can be reduced to a single phonon mode description, consistent with other approaches to the phenomenon of inelastic tunneling in heterostructures. A unifying connection is made with the phenomena of resonant electron scattering through molecules in the gas phase and of tunneling of electrons through adsorbates in scanning tunneling microscopy.