The OTELO survey. II. The faint-end of the H$\alpha$ luminosity function
  at z $\sim$ 0.40
  We take advantage of the capability of the OTELO survey to obtain the
H$\alpha$ luminosity function (LF) at ${\rm z}\sim0.40$. Because of the deepest
coverage of OTELO, we are able to determine the faint end of the LF, and thus
better constrain the star formation rate and the number of galaxies at low
luminosities. The AGN contribution to this LF is estimated as well. We make use
of the multi-wavelength catalogue of objects in the field compiled by the OTELO
survey, which is unique in terms of minimum flux and equivalent width. We also
take advantage of the pseudo-spectra built for each source, which allow the
identification of emission lines and the discrimination of different types of
objects. The H$\alpha$ luminosity function at $z\sim0.40$ is obtained, which
extends the current faint end by almost 1 dex, reaching minimal luminosities of
$\log_{10}L_{\rm lim}=38.5$ erg s$^{-1}$ (or $\sim0.002\, \text{M}_\odot\text{
yr}^{-1})$. The AGN contribution to the total H$\alpha$ luminosity is
estimated. We find that no AGN should be expected below a luminosity of
$\log_{10}L=38.6$ erg s$^{-1}$. From the sample of non-AGN (presumably, pure
SFG) at $z\sim0.40$ we estimated a star formation rate density of $\rho_{\rm
SFR}=0.012\pm0.005\ {\rm \text{M}_{\odot}\ yr^{-1}\ Mpc^{-3}}$.
