Several potentials appear to permit unexpected low-energy quantum tunneling in both Schrodinger and Dirac equations.
A very recent preprint in the Schrodinger case --
"The paradoxical zero reflection at zero energy"
http://arxiv.org/pdf/1605.07315v2.pdf
A paper examining the phenomenon for the Dirac Equation is --
"Low Momentum Scattering in the Dirac Equation"
http://arxiv.org/pdf/hep-th/0112156.pdf
A web search reveals many more. Most, but not all, examine the 1-dimensional case.
It would be interesting to know whether these are just mathematical curiosities, or can have real, measurable effects.