Symmetry in Nonlinear Mathematical Physics - 2009

Roderick Melnik (Wilfrid Laurier University, Waterloo, Canada)

Multiband Hamiltonians of the Luttinger–Kohn theory and ellipticity requirements

Symmetry in a wide range of mathematical models based on nonlinear PDEs has profound influence on the development of tools for the analysis of such models. While such tools have been developed extensively for single PDEs, especially of elliptic type, realistic problems require dealing with coupled systems of PDEs. In this contribution, we focus on one such system that is widely used for the description of low dimensional quantum nanostructures such as quantum dots, namely on the system of PDEs derived from the Luttinger–Kohn effective mass theory. One of the long-standing problems in this theory for multiband Hamiltonians is the appearance of spurious solutions [Commun. Comput. Phys. 6 (2009), 699–729]. This problem is closely related to the violation of the ellipticity of the underlying system, the fact that until recently has been largely overlooked. We demonstrate that in many cases the model derived by a formal application of the Luttinger–Kohn theory needs to be modified to satisfy the ellipticity requirement and we show a constructive way to achieve that based on simple geometrical interpretations.

This is a joint work with Dima Sytnyk and Sunil Patil.