  We study the generalized finite element methods (GFEMs) for the second-order
elliptic eigenvalue problem with an interface in 1D. The linear stable
generalized finite element methods (SGFEM) were recently developed for the
elliptic source problem with interfaces. We first generalize SGFEM to arbitrary
order elements and establish the optimal error convergence of the approximate
solutions for the elliptic source problem with an interface. We then apply the
abstract theory of spectral approximation of compact operators to establish the
error estimation for the eigenvalue problem with an interface. The error
estimations on eigenpairs strongly depend on the estimation of the discrete
solution operator for the source problem. We verify our theoretical findings in
various numerical examples including both source and eigenvalue problems.
