  We consider the propagation of a neutrino or an antineutrino in a medium
composed of fermions $f$ and scalars $\phi$ interacting via a Yukawa-type
coupling of the form $\bar f\nu\phi$, for neutrino energies at which the
processes like $\nu + \phi \leftrightarrow f$ or $\nu + \bar f \leftrightarrow
\bar\phi$, and the corresponding ones for the antineutrino, are kinematically
accessible. The relevant energy values are around $|m^2_\phi - m^2_f|/2m_\phi$
or $|m^2_\phi - m^2_f|/2m_f$, where $m_\phi$ and $m_f$ are the masses of $\phi$
and $f$, respectively. We refer to either one of these regions as a resonance
energy range. Near these points, the one-loop formula for the neutrino
self-energy has a singularity. From a technical point of view, that feature is
indicative that the self-energy acquires an imaginary part, which is associated
with damping effects and cannot be neglected, while the integral formula for
the real part must be evaluated using the principal value of the integral. We
carry out the calculations explicitly for some cases that allow us to give
analytic results. Writing the dispersion relation in the form $\omega = \kappa
+ V_{\text{eff}} - i\gamma/2$, we give the explicit formula for
$V_{\text{eff}}$ and $\gamma$ for the cases considered. When the neutrino
energy is either much larger or much smaller than the resonance energy,
$V_{\text{eff}}$ reduces to the effective potential that has been already
determined in the literature in the high or low momentum regime, respectively.
The virtue of the formula we give for $V_{\text{eff}}$ is that it is valid also
in the \emph{resonance energy range}, which is outside the two limits
mentioned. As a guide to possible applications we give the relevant formulas
for $V_{\text{eff}}$ and $\gamma$, and consider the solution to the oscillation
equations including the damping term, in a simple two-generation case.
