Search for dark energy potentials in quintessence theory
  The time evolution of the equation of state $w$ for quintessence models 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
mixed ( $V=M^{4+\gamma}\exp{(\beta M/Q)}/Q^{\gamma}$), 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$.
  From these specific potentials, we can predict the $ n+1 $ and higher
derivative of $w$ ; $ dw^{n+1}/da^{n+1}, \cdots$.
  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.
  At least it will satisfy the necessary conditions.
  Numerical analysis for $d^3w/da^3$ are made under some specified parameters
in the investigated potentials, except the mixed one.
  It becomes possible to distinguish the potentials by the accurate observing
$dw/da$ and $d^2w/da^2$ in some parameters.
