1994/04/29 by E. H. Lieb, Élliott H. Lieb, Jan Philip Solovej +6
Chemistry · Mathematics · Physics and Astronomy · #Advanced Chemical Physics Studies #Advanced Physical and Chemical Molecular Interactions #Condensed Matter (cond-mat) #FOS: Physical sciences #Spectral Theory in Mathematical Physics #cond-mat
paper · pdf · doi:10.48550/arxiv.cond-mat/9404099
20 pages, self-extracting uufile containing a plain TeX file and 2 postscript figures
arxiv created 1994/04/29 · openalex publication_date 1994/04/29 · arxiv updated 2009/11/30 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
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.