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The energy asymptotics of large Coulomb systems. (English) Zbl 0834.47066

Balslev, Erik (ed.), Schrödinger operators. The quantum mechanical many-body problem. Proceedings of a workshop, held at Aarhus, Denmark, 15 May - 1 August, 1991. Berlin: Springer-Verlag. Lect. Notes Phys. 403, 79-99 (1992).
Consider a molecule consisting of \(N\) quantized electrons at positions \(x_i\), and \(M\) nuclei of charges \(Z= (Z_1,\dots, Z_M)\) fixed at positions \(y= (y_1,\dots, y_M)\). The Schrödinger Hamiltonian of such a system is given by \[ H_{Z,N}= \sum^N_{i= 1} (- \Delta_{x_i}+ V_{\text{Coulomb}}(x_i))+ {1\over 2} \sum_{i\neq j} {1\over |x_i- x_j|},\;V_{\text{Coulomb}}(x)= - \sum^M_{j= 1} {Z_j\over |x- y_j|} \] acting on \({\mathcal H}= \bigwedge^N_{i= 1} L^2(\mathbb{R}^3\times \mathbb{Z}_q)\); in this exposition, in order to simplify notation, we neglect spin by putting \(q= 1\). Define the ground state of such a system by \[ E(Z, y)= \inf_N E(Z, y; N),\quad E(Z, y; N)= \inf_{\begin{smallmatrix} |\psi|= 1\\ \psi\in {\mathcal H}\end{smallmatrix}} \langle H_{Z, N} \psi, \psi\rangle. \] When \(M= 1\), this system is an atom. In this case, we can assume \(y= 0\) and we denote its energy simply by \(E_{\text{atom}}(Z)\). It is a remarkable fact that \(E(Z)\) behaves in a very simple way when \(Z\to \infty\), as follows:
Theorem 1: \[ \begin{aligned} E(Z, y; |Z|) & = - c_0 Z^{7/3}+ \textstyle{{1\over 8}} Z^2+ O(Z^{2- a})\qquad a> 0,\\ E_{\text{atom}}(Z) & = - c_0 Z^{7/3}+ \textstyle{{1\over 8}} Z^2+ O(Z^{{5\over 3}- {1\over 2835}}).\end{aligned} \] In the molecular case, the formula holds for all \(y\) such that \(|y_j- y_k|> Z^{- {2\over 3}+ \varepsilon}\), and for \(Z\to \infty\) along a given direction.
The purpose of this paper is to give a brief joint expository presentation of the proof of Theorem 1.
For the entire collection see [Zbl 0771.00018].

MSC:

47N50 Applications of operator theory in the physical sciences
81V10 Electromagnetic interaction; quantum electrodynamics
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