Overlapping Schwarz method is one of the most important methods for computing the large-scale discrete problems arising from partial differential equations. The previous proved bound is C(1+H^2/δ^2) for the condition number of the overlapping domain decomposition methods in H(curl; Ω), where H and δ are the sizes of subdomains and overlaps respectively. But all numerical results indicate that the best bound is C(1+H/δ). In this talk, we shall solve this long-standing open problem by proving that is indeed the best bound. Based on the overlapping Schwarz methods, we shall propose a two–level preconditioned Helmholtz subspace iterative (PHSI) method for solving algebraic eigenvalue problems resulting from edge element approximation of Maxwell eigenvalue problems. The two-level PHSI method may compute simple eigenpairs, multiple eigenpairs and clustered eigenpairs. This is a joint work with Prof. Xuejun Xu and Prof. Shangyou Zhang.