The FEM for a Schroedinger Equation

Hengguang Li
Penn State University

Abstract: Associated to different potentials (usually singular functions), Schroedinger type equations in quantum mechanics describe fundamental physical quantities of particles, while rigorous mathematical analysis of their solutions and effective numerical solvers are rarely found in the literature, due to various difficulties from singular potentials in the differential operator. This talk discusses a priori estimates of the Schroedinger type equation with the centrifugal potential in a class of weighted Sobolev spaces. Based on the theoretical analysis, we show some new applications of the finite element method on this type of Schroedinger equations that contain singular coefficients.

In particular, the well-posedness, regularity and Fredholm property of the solution are established in weighted Sobolev spaces, which motivates constructions of special finite element subspaces to cover the quasi-optimal rate of convergence for the finite element solution. In addition, a regularity estimate in these spaces allows us to extend current results to more general Schroedinger type operators.