  We study the existence of positive solution for the one dimensional
Schr\"odinger equation with mixed Lioville-Weyl fractional derivatives
\begin{eqnarray*}\label{Eq00}
_{t}D_{\infty}^{\alpha}({_{-\infty}}D_{t}^{\alpha}u(t)) + V(t) u(t) = &
f(u(t)),\;\;t\in \mathbb{R}\\ u\in H^{\alpha}(\mathbb{R}).\nonumber
\end{eqnarray*} Furthermore, we analyse radial symmetry property of these
solutions. The proof is carried out by using variational methods jointly with
comparison and rearrangement argument.
