Perturbations of the Dirichlet Problem and Error Bounds

Abstract

The Dirichlet problem on a bounded planar domain is more readily understood and solved for the Laplace operator than it is for a Schrodinger operator. When the potential function is small, we might hope to approximate the solution to the Schrodinger equation with the solution to the Laplace equation. In this vein we develop a series expansion for the solution and give explicit bounds on the error terms when truncating the series. We also examine a handful of examples and derive similar results for the Green function and Dirichlet-to-Neumann map.