Existence and concentration of solution for a fractional Hamiltonian systems with positive semi-definite matrix

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$isaparameter,$L\in C($\mathbb{R}$,$\mathbb{R}$^{n^2})$isasymmetricmatrixforall$t\in $\mathbb{R}$$,$W\in C^1($\mathbb{R}$ \times $\mathbb{R}$^n,$\mathbb{R}$)$. Assuming that $L(t)$ is a positive semi-definite symmetric matrix for all $t\in $\mathbb{R}$$,thatis,$L(t)\equiv 0$ 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 $\lambda \to \mathrm{\infty }$, and converges to some $\tilde{u}\in H^{\alpha }(\mathbb{R},{\mathbb{R}}^n)$. 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 $\alpha =1$.