In this talk, we propose and analyze the elliptic eigenvalue problems by using the weak Galerkin method. In contrast to the conforming finite element method, the lower bounds of eigenvalues are considered. We prove that the weak Galerkin method produces asymptotic lower bounds by using the high order polynomials, and produces guaranteed lower bounds by using the lowest order polynomials. Some numerical acceleration techniques are also applied to the weak Galerkin method, and the numerical experiments are presented to verify the theoretical analysis.