  We present here a detailed pulsational study applied to low-mass He-core
white dwarfs, based on full evolutionary models representative of these
objects. The background stellar models on which our pulsational analysis was
carried out were derived by taking into account the complete evolutionary
history of the progenitor stars, with special emphasis on the diffusion
processes acting during the white dwarf cooling phase. We computed nonradial
$g$-modes to assess the dependence of the pulsational properties of these
objects with stellar parameters such as the stellar mass and the effective
temperature, and also with element diffusion processes. We also performed a g-
and p-mode pulsational stability analysis on our models and found well-defined
blue edges of the instability domain, where these stars should start to exhibit
pulsations. We found substantial differences in the seismic properties of white
dwarfs with $M_* \gtrsim 0.20 M_{\odot}$ and the extremely low-mass (ELM) white
dwarfs ($M_* \lesssim 0.20 M_{\odot}$). Specifically, $g$-mode pulsation modes
in ELM white dwarfs mainly probe the core regions and are not dramatically
affected by mode-trapping effects by the He/H interface, whereas the opposite
is true for more massive He-core white dwarfs. We found that element diffusion
processes substantially affects the shape of the He/H chemical transition
region, leading to non-negligible changes in the period spectrum of low-mass
white dwarfs. Our stability analysis successfully predicts the pulsations of
the only known variable low-mass white dwarf (SDSS J184037.78+642312.3), and
also predicts both $g$- and $p$-mode pulsational instabilities in a significant
number of known low-mass and ELM white dwarfs.
