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Asymptotics of the ground state energies of large Coulomb systems. (English) Zbl 0789.35135

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).
Reviewer: B.Helffer (Paris)

MSC:

35Q40 PDEs in connection with quantum mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81V55 Molecular physics
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