Regularity and hp discontinuous Galerkin finite element approximation of linear elliptic eigenvalue problems with singular potentials

We study the regularity in weighted Sobolev spaces of Schrodinger-type eigenvalue problems, and we analyze their approximation via a discontinuous Galerkin (dG) hp finite element method. In particular, we show that, for a class of singular potentials, the eigen-functions of the operator belong to analytic-type non-homogeneous weighted Sobolev spaces. Using this result, we prove that an isotropically graded hp dG method is spectrally accurate, and that the numerical approximation converges with exponential rate to the exact solution. Numerical tests in two and three dimensions confirm the theoretical results and provide an insight into the behavior of the method for varying discretization parameters.