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The transmission coefficient for passage through a barrier with a parabolic maximum is computed rigorously when linear passive dissipation is present. The exact result for energies above and below the barrier height can be expressed in terms of the Coleman renormalized one-bounce time. The quasiclassical (WKB) approximation is recovered for energies significantly below the barrier height where "friction" enhances the tunneling probability.
Widom et al. (Mon,) studied this question.