Quest for potentials in the quintessence scenario
  The time variation of the equation of state $w$ for quintessence scenario
with a scalar field as dark energy is studied up to the third derivative
($d^3w/da^3$) with respect to the scale factor $a$, in order to predict the
future observations and specify the scalar potential parameters with the
observables. The third derivative of $w$ for general potential $V$ is derived
and applied to several types of potentials. They are the inverse power-law
($V=M^{4+\alpha}/Q^{\alpha}$), the exponential ($V=M^4\exp{(\beta M/Q)}$), the
cosine ($V=M^4(\cos (Q/f)+1)$) and the Gaussian types
($V=M^4\exp(-Q^2/\sigma^2)$), which are prototypical potentials for the
freezing and thawing models.
  If the parameter number for a potential form is $ n$, it is necessary to find
at least for $n+2$ independent observations to identify the potential form and
the evolution of the scalar field ($Q$ and $ \dot{Q} $).
  Such observations would be the values of $ \Omega_Q, w, dw/da. \cdots $, and
$ dw^n/da^n$.
  Since four of the above mentioned potentials have two parameters, it is
necessary to calculate the third derivative of $w$ for them to estimate the
predict values.
  If they are tested observationally, it will be understood whether the dark
energy could be described by the scalar field with this potential.
  Numerical analysis for $d^3w/da^3$ are made under some specified parameters
in the investigated potentials.
  It becomes possible to distinguish the freezing and thawing modes by the
accurate observing $dw/da$ and $d^2w/da^2$ in some parameters.
