On sharp lower bound of the spectral gap for a Schr\"odinger operator and some related results

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.