Expectations and sensitivity of the scalar field Dark Energy
  reconstruction from the SNe Ia data
  The current paper is addressing the possibility of the Dark Energy scalar
field potential reconstruction from the SNe Ia data and the problems arising
during the process. We describe the method and test its limits, stability of
the reconstruction with respect to the $\Omega_m$ and $H_0$ parameters, as well
as several issues connected to the errors propagation, with use of synthetic
data. After that, we test the method with real Union2.1 and JLA, as well as
recent PANTHEON SNe Ia datasets. It worths mentioning that in our approach we
assume no {\it Ans\"atzen} on the dynamical variables (e.g., $H(z)$), which
makes our method free of any degeneracies and biases which arise if one assume
one or another way to parameterize $H(z)$, or the equation of state, or some
other variable. On the other hand, the price we pay for this freedom is immense
-- although the scheme demonstrates perfect reconstruction in the case of
synthetic data, the reconstruction from the real data seems almost impossible,
at least on the current data precision level. The errors of the resulting
potential and of the kinetic term are huge; for the Hubble parameter they are
smaller but still severalfold larger than those obtained with parameterization.
Finally, we test the method with non-SNe $H(z)$ datasets, and the results are
also disappointing -- similar to the SNe case, the reconstruction is not
possible as the reconstructed kinetic term enters the negative values. This
could be treated as a manifestation of the insufficient data precision, or as
the overestimation of $(H_0, \Omega_m)$ values -- we discuss both
possibilities.
