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.