We are concerned with the existence of ground states solutions to the following fractional Hamiltonian systems:[Formula presented] where α∈(1/2,1)α∈(1/2,1), t∈ℝt∈ℝ, u∈ℝnu∈ℝn, λ>0λ>0 is a parameter, L∈C(ℝ,ℝn2)L∈C(ℝ,ℝn2) is a symmetric matrix for all t∈ℝt∈ℝ, W∈C1(ℝ×ℝn,ℝ)W∈C1(ℝ×ℝn,ℝ) and ∇W(t,u)∇W(t,u) is the gradient of W(t,u)W(t,u) at uu. Assuming that L(t)L(t) is a positive semi-definite symmetric matrix for all t∈ℝt∈ℝ, that is, L(t)≡0L(t)≡0 is allowed to occur in some finite interval TT of ℝℝ, W(t,u)W(t,u) satisfies the Ambrosetti-ℝabinowitz condition and some other reasonable hypotheses, we show that (FHS)λλ has a ground sate solution which vanishes on ℝ∖Tℝ∖T as λ→∞λ→∞, and converges to u∈Hα(ℝ,ℝn)u∈Hα(ℝ,ℝn), where u∈Eα0u∈E0α is a ground state solution of the Dirichlet BVP for fractional systems on the finite interval TT. Recent results are generalized and significantly improved.