Lieb E. H. , Solovej J. P. , Yngvason J.
Quantum dots
(95K, Plain TeX)
ABSTRACT. 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.