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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 \({\mathbf H}\geq E_{TH}(1,N/{\mathbf Z}){\mathbf Z}^{7/3}+(q/8){\mathbf Z}^ 2-\text{Const.} {\mathbf Z}^{53/27}\).

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.)
Zbl 0247.47018
Full Text: DOI Numdam EuDML
[1] V. BACH , A Proof of Scott’s Conjecture for Ions . [Rep. Math. Phys., (to appear).]. · Zbl 0732.58042
[2] F. A. BEREZIN , Wick and Anti-Wick Operator Symbols . (Math. U.S.S.R. Sbornik, Vol. 15, 1971 , pp. 578-610). MR 45 #929 | Zbl 0247.47018 · Zbl 0247.47018
[3] W. HUGHES , An Atomic Energy Lower Bound that Gives Scott’s Correction . (Ph. D. thesis, Princeton, Department of Mathematics, 1986 ).
[4] W. HUGHES , An Atomic Lower Bound that Agrees with Scott’s Correction . (Advances in Mathematics, 1990 , pp. 213-270). MR 91c:81180 | Zbl 0715.46046 · Zbl 0715.46046
[5] E. H. LIEB , Thomas-Fermi and Related Theories of Atoms and Molecules (Rev. Mod. Phys., 53, 1981 , pp. 603-604). MR 83a:81080a | Zbl 1049.81679 · Zbl 1049.81679
[6] E. H. LIEB and B. SIMON , The Thomas-Fermi Theory of Atoms, Molecules and Solids . (Adv. Math., Vol. 23, 1977 , pp. 22-116). MR 55 #1964 | Zbl 0938.81568 · Zbl 0938.81568
[7] E. H. LIEB and W. E. THIRRING , Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and their Relation to Sobolev Inequalities , in E. H. LIEB, B. SIMON and A. S. WIGHTMAN Ed., Studies in Mathematical Physics : Essays in Honor of Valentine Bargmann , Princeton University Press, Princeton, 1976 . Zbl 0342.35044 · Zbl 0342.35044
[8] H. SIEDENTOP and R. WEIKARD , On Some Basic Properties of Density Functionals for Angular Momentum Channels . (Rep. Math. Phys., Vol. 28, 1986 , pp. 193-218). MR 90a:81196 | Zbl 0644.46059 · Zbl 0644.46059
[9] H. SIEDENTOP and R. WEIKARD , On the Leading Correction of the Thomas-Fermi Model : Lower Bound - with an Appendix by A. M. K. Müller . (Invent. Math., 97, 1989 , pp. 159-193). MR 90k:81285 | Zbl 0689.34011 · Zbl 0689.34011
[10] H. SIEDENTOP and R. WEIKARD , On the Leading Energy Correction for the Statistical Model of the Atom : Interacting Case . (Commun. Math. Phys., Vol. 112, 1987 , pp. 471-490). Article | MR 89a:81022 | Zbl 0920.35120 · Zbl 0920.35120
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