Analytical treatment of Schrödinger problems with geometrical perturbations

Abstract

An analytical theory for calculating shifts in eigenvalues to Schrödinger problems due to geometrical perturbations is proposed. The possibility of growing practically any nano-heterostructure today makes it interesting to examine the influence of shape and size on, e.g., electronic, optical and magnetic properties. For that, detailed knowledge of shifts in eigenvalues induced by changes in geometry, either intentionally or unintentionally, is important. Several examples are presented so as (a) to verify the geometry-perturbative model against exact calculations and (b) to attack problems which cannot be solved exactly without resorting to numerical partial differential equation (PDE) techniques. The latter include the cases of a particle confined to a three-dimensional cylinder with a sinusoidally varying radius as a function of the axial coordinate in the presence (or absence) of a dc electric field along the cylinder axis.