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A new phase space localization technique with application to the sum of negative eigenvalues of Schrödinger operators. (English) Zbl 0762.47022
The Hamiltonian \({\mathbf H}\) of \(N\) electrons in the field of nucleus of charge \({\mathbf Z}\) with selfadjoint realization in \(\bigwedge_{i=1}^ N (L^ 2(R^ 3)\otimes C^ q)\) is \[ {\mathbf H}=\sum_{i=1}^ N (- \Delta_ i-{\mathbf Z}/| r_ i|)+\sum_{i,j=1, i<j}^ N (1/| r_ i-r_ j|), \] where \(q\) is the number of spin states of a single electron. It is known that \[ E_ Q({\mathbf Z},N)=E_{TF}(1,N/{\mathbf Z}){\mathbf Z}^{7/3}+(q/8){\mathbf Z}^ 2+O({\mathbf Z}^{47/24}) \] for fixed or negative degree of ionization \(\lambda=1-N/{\mathbf Z}\) where \(E_ Q({\mathbf Z},N)=\inf\sigma({\mathbf H})\) and \(E_{TF}({\mathbf Z},N)\) is the Thomes-Fermi energy.
This paper gives a simplified proof of the lower bound on \(E_ Q({\mathbf Z},N)\) using Macke orbitals that require “almost only” known properties from the upper bound. The Macke orbitals yield a phase space localization through their densities \(\rho_ \ell\) and momenta \(\pi k_{\ell,n}\).
The proof is somewhat along the lines of F. A. Berezin [Math. U.S.S.R., Sbornik, 15, 577-606 (1971; Zbl 0247.47018)] and E. H. Lieb [Rev. Mod. Phys. 53, 603-604 (1981)]. The author derives the following Scott type lower bound \({\mathbf H}\geq E_{TH}(1,N/{\mathbf Z}){\mathbf Z}^{7/3}+(q/8){\mathbf Z}^ 2-\text{Const.} {\mathbf Z}^{53/27}\).

MSC:
47F05 General theory of partial differential operators
47N50 Applications of operator theory in the physical sciences
47E05 General theory of ordinary differential operators
35J10 Schrödinger operator, Schrödinger equation
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
Citations:
Zbl 0247.47018
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References:
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