In this paper, we study the one dimensional Schrodinger equation with Liouville–Weyl fractional derivatives $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} {}_tD_\infty ^\alpha ({}_{-\infty }D_t^\alpha u(t))+V(t)u(t)=f(t,u(t)), \ \ \ \ t\in {\mathbb {R}},\\ u \in {H^\alpha }(\mathbb {R}), \end{array} \right. \end{aligned}$$ For the case when f(t, u) is superlinear at infinity, by applying Ekeland Variational Principle and Mountain Pass Theorem, we obtain the existence of at least two nontrivial solutions. For the case of asymptotically linear at infinity, we prove that the above problem has one nontrivial solution under certain assumptions. Some recent results are extended and improved.