Quantum Dots, in

Atomic-like systems in which electronic motion is two dimensional are now realizable as " quantum dots ". In place of the attraction of a nucleus there is a confining potential, usually assumed to be quadratic. Additionally, a perpendicular magnetic field B may be present. We review some recent rigorous results for these systems. We have shown that a Thomas-Fermi type theory for the ground state is asymptotically correct when N and B tend to infinity. There are several mathematically and physically novel features. 1. The derivation of the appropriate Lieb-Thirring inequality requires some added effort. 2. When B is appropriately large the TF " kinetic energy " term disappears and a peculiar " classical " continuum electrostatic theory emerges. This is a two dimensional problem, but with a three dimensional Coulomb potential. 3. Corresponding to this continuum theory is a discrete " classical " electrostatic theory. The former provides an upper bound and the latter a lower bound to the true quantum energy; the problem of relating the two classical energies offers an amusing exercise in electrostatics. c 1993 in image and content by the authors. Reproduction of this article, by any means, is permitted for non-commercial purposes.