  We present the first systematic work for deriving a posteriori error
estimates for general non-polynomial basis functions in an interior penalty
discontinuous Galerkin (DG) formulation for solving eigenvalue problems
associated with second order linear operators. Eigenvalue problems of such
types play important roles in scientific and engineering applications,
particularly in theoretical chemistry, solid state physics and material
science. Based on the framework developed in [{\it L. Lin, B. Stamm,
http://dx.doi.org/10.1051/m2an/2015069}] for second order PDEs, we develop
residual type upper and lower bound error estimates for measuring the a
posteriori error for eigenvalue problems. The main merit of our method is that
the method is parameter-free, in the sense that all but one solution-dependent
constants appearing in the upper and lower bound estimates are explicitly
computable by solving local and independent eigenvalue problems, and the only
non-computable constant can be reasonably approximated by a computable one
without affecting the overall effectiveness of the estimates in practice.
Compared to the PDE case, we find that a posteriori error estimators for
eigenvalue problems must neglect certain terms, which involves explicitly the
exact eigenvalues or eigenfunctions that are not accessible in numerical
simulations. We define such terms carefully, and justify numerically that the
neglected terms are indeed numerically high order terms compared to the
computable estimators. Numerical results for a variety of problems in 1D and 2D
demonstrate that both the upper bound and lower bound are effective for
measuring the error of eigenvalues and eigenfunctions.
