  In a recent article (arxiv:1803.06332) we noticed that the electron density
in condensed matter exhibits large spikes close to the atomic nuclei. We showed
that these spikes in the electron densities, 3-4 orders of magnitude larger
than those inside the Sun's core, have no effect on the neutrino emission and
absorption probabilities or on the neutrinoless double beta decay probability.
However, it was not clear if the effect of these spikes is equivalent to that
of an average constant electron density in matter. We investigated these
effects by a direct integration of the coupled Dirac equations describing the
propagation of flavor neutrinos into, through, and out of the matter. We found
little evidence that these spikes affect the standard oscillations
probabilities, but found a new fast and efficient algorithm of calculating
these probabilities for neutrinos propagating through varying electron
densities.
