Context : In recent years , many low-mass ( \lesssim 0.45 M _ { \odot } ) white dwarf stars expected to harbor He cores have been detected in the field of the Milky Way and in several galactic globular and open clusters .
Until recently , no objects of this kind showed pulsations .
This situation has changed recently with the exciting discovery of SDSS J184037.78+642312.3 , the first pulsating low-mass white dwarf star .

Aims : Motivated by this extremely important finding , and in view of the very valuable asteroseismological potential of these objects , we present here a detailed pulsational study applied to low-mass He-core white dwarfs with masses ranging from 0.17 to 0.46 M _ { \odot } , based on full evolutionary models representative of these objects .
This study is aimed to provide a theoretical basis from which to interpret future observations of variable low-mass white dwarfs .

Methods : 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 .

Results : 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 , in particular in the range of stellar masses characteristic of ELM objects .
Finally , our stability analysis successfully predicts the pulsations of the only known variable low-mass white dwarf ( SDSS J184037.78+642312.3 ) at the right effective temperature , stellar mass and range of periods .

Conclusions : Our computations predict both g - and p -mode pulsational instabilities in a significant number of known low-mass and ELM white dwarfs .
It is worth observing these stars in order to discover if they pulsate .