Asymptotics of the ground state energies of large Coulomb systems
type: Item
instance of: scholarly article
title: Asymptotics of the ground state energies of large Coulomb systems
zbMATH Open document ID: 0789.35135
DOI: 10.2307/2946613
published in: Annals of Mathematics. Second Series
publication date: 1994-06-16T00:00:00Z
review text: The authors consider a molecule consisting of \(N\) electrons, each of charge \(-1\), and \(M\) nuclei of charges \(Z_ 1,\dots, Z_ m\) s.t. \(N=\sum_{i=1}^ M Z_ i\). They assume that the nuclei are infinitely heavy and located at positions \(R_ 1, \dots,R_ M\). The Schrödinger operator of such a system is: \[ H(Z,R)= \sum_{i=1}^ N (-(1/2) \Delta_ i- V_ Z(x_ i,R))+ \sum_{i<j} | x_ i-x_ j|^{-1} \] acting on the fermionic space \(\Lambda_{i=1}^ N L^ 2(\mathbb{R}^ 3\times \mathbb{Z}_ 2)\). Here \(x_ i\in\mathbb{R}_ 3\) is the coordinate of the \(i\)th electron, \(\Delta_ i\) is the Laplacian in \(x_ i\), \(Z= (Z_ 1,\dots, Z_ M)\), \(R= (R_ 1,\dots, R_ M)\) and \(V_ Z(x,R)= \sum_{i=1}^ M Z_ i/ (| x-R_ i|)\). The article is devoted to the study of the ground state energy \(E(Z,R)\) of \(H(Z,R)\) as \(Z\to\infty\) along some direction in \(\mathbb{R}^ M\). The principal result is the proof of the Scott's conjecture for molecules that is the existence of the following asymptotics: \[ E(Z,R)= E^{TF}(Z,R)+ (1/2) \sum_ j Z_ j^ 2+ O_ \delta(a^{-1} | Z|^{(16/9+\delta)}) \] for any \(\delta>0\), where \(a=\min(| Z|^{1/3} | R_ i-R_ j|\), \(k\neq j;1)\). \(E^{TF}(Z,R)\) is the Thomas-Fermi energy which appears also as the Weylian term in the semiclassical approach of this spectral problem. The Scott's conjecture was only proved before for the case of atoms (see Hughes, Siedentop- Weikard, and improvements due to Feffermann-Seco).
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81Q20
Mathematics Subject Classification ID: 81V55
zbMATH DE Number: 490544
zbMATH Keywords: proof of the Scott's conjecture for molecules
author: Victor Ivrii
author: I. M. Sigal
reviewed by: Bernard Helffer
MaRDI profile type: Publication
OpenAlex ID: W2063301089
description: scientific article; zbMATH DE number 490544