  We study the existence of solutions for the following fractional Hamiltonian
systems $$ \left\{
  \begin{array}{ll}
  - _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla
W(t,u(t))=0,\\[0.1cm]
  u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n),
  \end{array} \right.
  \eqno(\mbox{FHS})_\lambda $$ where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$,
$u\in \mathbb{R}^n$, $\lambda>0$ is a parameter, $L\in
C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix for all $t\in
\mathbb{R}$, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$. Assuming
that $L(t)$ is a positive semi-definite symmetric matrix for all $t\in
\mathbb{R}$, that is, $L(t)\equiv 0$ is allowed to occur in some finite
interval $T$ of $\mathbb{R}$, $W(t,u)$ satisfies some superquadratic conditions
weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)$_\lambda$ has a
solution which vanishes on $\mathbb{R}\setminus T$ as $\lambda \to \infty$, and
converges to some $\tilde{u}\in H^{\alpha}(\R, \R^n)$. Here, $\tilde{u}\in
E_{0}^{\alpha}$ is a solution of the Dirichlet BVP for fractional systems on
the finite interval $T$. Our results are new and improve recent results in the
literature even in the case $\alpha =1$.
