Constraints for the thawing and freezing potentials
  We study the accelerating present universe in terms of the time evolution of
the equation of state $w(z)$ (redshift $z$) due to thawing and freezing scalar
potentials in the quintessence model. The values of $dw/da$ and $d^2w/da^2$ at
scale factor of $a = 1$ are associated with two parameters of each potential.
For five type scalar potentials, the scalar fields $Q$ and $w$ as function of
time $t$ and/or $z$ are numerically calculated under the fixed boundary
condition of $w(z=0)=-1+\Delta$. The observational constraint $w_{obs}$ (Planck
2015) is imposed to test whether the numerical $w(z)$ is in $w_{obs}$. Some
solutions show the thawing features in the freezing potentials. Mutually
exclusive allowed regions in $dw/da$ vs. $d^2w/da^2$ diagram are obtained in
order to identify the likely scalar potential and even the potential parameters
by future observational tests.
