  In this paper, we give an easy proof of the main results of Andrews and
Clutterbuck's paper [J. Amer. Math. Soc. 24 (2011), no. 3, 899--916], which
gives both a sharp lower bound for the spectral gap of a Schr\"oinger operator
and a sharp modulus of concavity for the logarithm of the corresponding first
eigenfunction. We arrive directly at same estimates by the `double coordinate'
approach and asymptotic behavior of parabolic flows. Although using the
techniques appeared in the above paper, we partly simplify the method and
argument. This maybe help to provide an easy way for estimating spectral gap.
Besides, we also get a new lower bound of spectral gap for a class of
Sch\"odinger operator.
