In this paper, we consider the following class of fractional Hamiltonian systems xD∞α(-∞Dxαu(x))+L(x)u(x)=∇W(x,u(x))+f(x),x∈Ru∈Hα(R,RN),where α∈ (1 / 2 , 1) , L∈C(R,RN2) is a symmetric positive definite matrix, W∈ C1(R× RN, R) is superquadratic and even in u. By using Bolle’s perturbation method in critical point theory, we prove the existence of infinitely many solutions in spite of the lack of the symmetry of this problem. Moreover, we study the spectral properties of the operator xD∞α(-∞Dxα)+L(x).