An elementary proof of the anti-monotonicity of the quantum mechanical  particle density with respect to the potential in the Hamiltonian is given  for a large class of admissible thermodynamic equilibrium distribution  functions. In particular the zero temperature case is included

Kaiser, Hans-Christoph

Neidhardt, Hagen

Rehberg, Joachim

English

An elementary proof of the anti-monotonicity of the quantum mechanical
particle density with respect to the potential in the Hamiltonian is given
for a large class of admissible thermodynamic equilibrium distribution
functions. In particular the zero temperature case is included

Repositorium für Naturwissenschaften und Technik  (TIB Hannover)

Monotonicity properties of the quantum mechanical particle density

Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics

Weierstraÿ-Institutfür Angewandte Analysis und Stohastikim Forshungsverbund Berlin e.V.Preprint ISSN 0946  8633Monotoniity properties of the quantum mehanialpartile densityDediated to Professor Hellmut BaumgärtelHans-Christoph Kaiser, Hagen Neidhardt, Joahim Rehbergsubmitted: 6th Deember 2007Weierstrass Institute for Applied Analysis and StohastisMohrenstr. 3910117 BerlinGermanyE-Mail: kaiserwias-berlin.deneidhardtwias-berlin.derehbergwias-berlin.deNo. 1275Berlin 20072000 Mathematis Subjet Classiation. 47H05, 47B10.Key words and phrases. Trae funtionals, trae lass operators, monotoniity.Edited byWeierstraÿ-Institut für Angewandte Analysis und Stohastik (WIAS)Mohrenstraÿe 3910117 BerlinGermanyFax: + 49 30 2044975E-Mail: preprintwias-berlin.deWorld Wide Web: http://www.wias-berlin.de/Monotoniity 1AbstratAn elementary proof of the anti-monotoniity of the quantum mehanialpartile density with respet to the potential in the Hamiltonian is given fora large lass of admissible thermodynami equilibrium distribution funtions.In partiular the zero temperature ase is inluded.1 IntrodutionThe Kohn-Sham system of Density Funtional Theory, see e.g. [15, 13, 3℄, has be-ome a widely used tool in the simulation of the eletroni struture of matter sineit was introdued by Kohn and Sham in 1965 [9℄. This was aknowledged by award-ing the Nobel prize for Chemistry in 1998 to Walter Kohn [8℄. On the other handthere are few mathematial investigation of the Kohn-Sham system up to now: Theinteresting question of bifuration of solutions has been treated in [14℄. For the Kohn-Sham system as a nonlinear system of partial dierential equations on a boundedspatial domain existene of solutions and a priori estimates have been proved, see [6℄,[7℄. The proof for the Kohn-Sham system with an exhangeorrelation term rests onthe existene (and regularity properties) of an unique solution for the orrespondingShrödingerPoisson system with only self-onsistent eletrostati interation. Theunderlying fundamental result is the anti-monotoniity of the quantum mehanialpartile density with respet to the potential in the Hamiltonian. This was observedin 1990 independently by Nier [12℄ and Caussigna et al. [1℄, and is losely relatedto the onvexity of von Neumann-type trae funtionals [11, hapter V.3℄, [10℄, [5℄.In this note we prove in an elementary way the anti-monotoniity of the quantummehanial partile density with respet to the potential in the Hamiltonian. Thus,we additionally enlarge the lass of admissible thermodynami equilibrium distribu-tion funtions used for the omposition of the quantum mehanial partile density.2 ResultsIn the following H is an innite dimensional, separable Hilbert spae with salarprodut (·, ·), and H is a self-adjoint operator on H with ompat resolvent.Theorem 2.1. Let U and V be bounded, self-adjoint operators on H. If f : R → RPreprint 1275, Weierstrass Institute for Applied Analysis and Stohastis, Berlin 20072 H.-Chr. Kaiser, H. Neidhardt, J. Rehbergis a funtion suh that f(H + U) and f(H + V ) are trae lass operators, thentr([f(H + U)− f(H + V )][U − V ])=∞∑k,l=1|(ψk, ζl)|2(f(λk)− f(µl))(λk − µl), (1)where {λk}∞k=1 and {µl}∞l=1 are the sequenes of eigenvalues (ounting multipliity)of the operators H + U and H + V , respetively, and {ψk}∞k=1, {ζl}∞l=1 are the or-responding sequenes of (normalised) eigenvetors.Remark 2.2. In ontrast to previous results (see e.g. [5℄) Theorem 2.1 does notrequire that the distribution funtion f is ontinuous or (Lebesgue) measurable. Thisis due to the fat that all ourring spetral measures E are disrete and, hene,all (salar) measures (E(·)ϕ, ϕ) are purely atomi. Consequently, every funtionf : R → R is measurable with respet to measures (E(·)ϕ, ϕ) with ϕ ∈ H, see e.g.[2, hapter 13.18℄. Thus, the integrals∫Rf(λ) d(E(λ)ϕ, ϕ) are well dened for allϕ ∈ H.In partiular, the harateristi funtion of the negative half axis is an admissibledistribution funtion. This ase orresponds to a quantum system at zero tempera-ture.Corollary 2.3. Under the assumptions of Theorem 2.1: If there are real numbersǫU and ǫV suh that the operators f(H + U − ǫU), and f(H + V − ǫV ) are nulearand, additionally, satisfytr(f(H + U − ǫU)) = tr(f(H + V − ǫV )), (2)thentr([f(H + U − ǫU )− f(H + V − ǫU)][U − V ])=∑∞k,l=1 |(ψk, ζl)|2 [f(λk − ǫU )− f(µl − ǫV )] [λk − ǫU − µl + ǫV ] . (3)These results have onsequenes for the quantum mehanial partile density. Letus assume that H is a semi-bounded, self-adjoint operator with ompat resolvent inL2(Ω), where Ω is a bounded Lipshitz domain in the up to three dimensional spaeRd. In partiular, H an be a Shrödinger operator. The operatorH is perturbed bya bounded, self-adjoint multipliation operator U indued by a real potential U fromL∞(Ω). Then H+U has also a ompat resolvent, and we denote by {λk(U)}∞k=1 and{ψk(U)}∞k=1 the sequenes of eigenvalues and (normalised) eigenfuntions of H + Uounting multipliity.N (U)def=∞∑k=1f(λk − ǫU)|ψk(U)|2(4)is the quantum mehanial partile density assoiated with the potential U withrespet to the free Hamiltonian H and the thermodynami equilibrium distributionPreprint 1275, Weierstrass Institute for Applied Analysis and Stohastis, Berlin 2007Monotoniity 3funtion f . If there is Fermi-Dira distribution, then, in the three-dimensional ase(d = 3) at positive temperature, f is the Fermi funtion f(s) = 1/(1 + exp(−s)). Ifthe sum∑∞k=1 f(λk − ǫU) onverges, then the (non-linear) partile density operatorN is a map from L∞(Ω) to L1(Ω). The normalising shift ǫU is the Fermi level denedbyN =∫ΩN (U)(x) dx =∞∑k=1f(λk − ǫU ), (5)where N is total number of partiles onned in Ω. If f is not stritly monotone,then the Fermi level is not neessarily uniquely determined by (5). However, we allany solution of (5) a Fermi level of the quantum system.Corollary 2.4. Let H be a semi-bounded, self-adjoint operator in L2(Ω) and {λk}∞k=1the sequene of its eigenvalues ounting multipliity. If the distribution funtionf : R → R+ is monotonously dereasing and obeys∞∑k=1|f(λk + α)| <∞ for any real number α, (6)then the partile density operator (4) is anti-monotone from L∞(Ω) to L1(Ω), i.e.∫Ω(N (U)(x)−N (V )(x))(U(x) − V (x)) dx ≤ 0 (7)for all real potentials U , V ∈ L∞(Ω).Remark 2.5. Corollary 2.4 an be extended to more general perturbations U , Vthan just bounded ones. If, e.g. H and f are hosen suh that f(H + W )X isnulear for all multipliation operators W and X indued by funtions from L2(Ω),then the (extended) mappingL2(Ω) ∋W → −N (W ) ∈ L2(Ω)is also monotone, see [6℄, [7℄, [5℄.3 ProofsFor the proof of our results we need the following simple fats on the summabil-ity of double sums∑∞k,l=1 akl. We say the double sum onverges if the sequene{An}∞n=1 of partial sums An =∑nk,l=1 akl onverges. If∑∞k=1∑∞l=1 |akl| onverges,then∑∞k,l=1 akl is summable, the sums∑∞k=1∑∞l=1 akl and∑∞l=1∑∞k=1 akl onverge,and∞∑k,l=1akl =∞∑k=1∞∑l=1akl,=∞∑l=1∞∑k=1akl, (8)see [4, Theorem XI.5.2℄.Preprint 1275, Weierstrass Institute for Applied Analysis and Stohastis, Berlin 20074 H.-Chr. Kaiser, H. Neidhardt, J. RehbergProof of Theorem 2.1Beause H is a self-adjoint operator with ompat resolvent in the separable Hilbertspae H the operators H + U and H + V also have a ompat resolvent. Sinef(H + U) and f(H + V ) are trae lass operators, there istr((f(H + U)− f(H + V )) (U − V ))= tr(f(H + U)(U − V ))− tr(f(H + V )(U − V )).The trae an be expressed in terms of an eigensystem:tr(f(H + U)(U − V ))=∞∑k=1(f(H + U)(U − V )ψk, ψk) =∞∑k=1f(λk)((U − V )ψk, ψk),where the sum on the right-hand side is absolutely onvergent. Further, one andevelop eah ψk into a Fourier series with respet to the system {ζl}∞l=1:ψk =∞∑l=1(ψk, ζl)ζl.Hene,tr(f(H + U)(U − V ))=∞∑k=1f(λk)((U − V )ψk, ψk) =∞∑k=1f(λk)∞∑l=1(ψk, ζl)((U − V )ψk, ζl).As the sum∑∞k=1 |f(λk)|∑∞l=1∣∣(ψk, ζl)((U − V )ψk, ζl)∣∣ onverges, (8) impliestr(f(H + U)(U − V )) =∞∑k,l=1f(λk)(ψk, ζl)((U − V )ψk, ζl).Next we note((U − V )ψk, ζl) = ([(H + U)− (H + V )]ψk, ζl) = (λk − µl)(ψk, ζl)for all k, l ∈ N. Hene,tr(f(H + U)(U − V )) =∞∑k,l=1f(λk)(ψk, ζl)(λk − µl)(ψk, ζl)=∞∑k,l=1f(λk)(λk − µl)|(ψk, ζl)|2. (9)Similarly one getstr(f(H + V )(U − V )) =∞∑k,l=1f(µl)(λk − µl)|(ψk, ζl)|2,whih, together with (9), implies (1).Preprint 1275, Weierstrass Institute for Applied Analysis and Stohastis, Berlin 2007Monotoniity 5Proof of Corollary 2.3From (2) one getstr (f(H + U − ǫU )− f(H + V − ǫV )) [ǫV − ǫU ] = 0,and onsequentlytr ([f(H + U − ǫU)− f(H + V − ǫV )][U − V ])= tr ([f(H + U − ǫU)− f(H + V − ǫV )][(U − ǫU)− (V − ǫV )]) .Now, Theorem 2.1, applied to U˜ = U − ǫU and V˜ = V − ǫV , implies the relation (3).Proof of Corollary 2.4One has to hek that for any U , V ∈ L∞ the relation (7) holds. First, one veriesthat f(H +W − ǫW ) is a trae lass operator for any bounded operator W . Indeed,there isH − ‖W‖ − |ǫW | ≤ H +W − ǫWin the sense of forms. Due to the minimax priniple, the monotone deay of thedistribution funtion f , and the supposition (6) the operator f(H + W − ǫW ) istrae lass. Thus, the series (4) onverges in L1(Ω). By straightforward alulationsone veries that the expression on the left hand side of (7) equals (3). As f dereasesmonotonously, the pre-fators(f(λk − ǫU )− f(µl − ǫV )) ((λk − ǫU)− (µl − ǫV ))are all non-positive, what proves the assertion.Referenes[1℄ Ph. Caussigna, B. Zimmermann, and R. Ferro. Finite element approximationof eletrostati potential in one dimensional multilayer strutures with quan-tized eletroni harge. Computing, 45:251264, 1990.[2℄ J. Dieudonné. Elements d'analyse 2. GauthierVillars, Paris, 1974.[3℄ R. M. Dreizler and E. K. U. Gross. Density Funtional Theory. Springer,Berlin, 1990.[4℄ G. M. Fihtenholz. Dierential- und Integralrehnung II (Dierential and in-tegral alulus II). Frankfurt am Main, 1990.[5℄ H.-Chr. Kaiser, H. Neidhardt, and J. Rehberg. Convexity of trae funtionalsand Shrödinger operators. J. Funtional Analysis, 234:4569, 2006.Preprint 1275, Weierstrass Institute for Applied Analysis and Stohastis, Berlin 20076 H.-Chr. Kaiser, H. Neidhardt, J. Rehberg[6℄ H.-Chr. Kaiser and J. Rehberg. About a one-dimensional stationaryShrödingerPoisson system with KohnSham potential. Zeitshrift für Ange-wandte Mathematik und Physik (ZAMP), 50:423458, 1999.[7℄ H.-Chr. Kaiser and J. Rehberg. About a stationary ShrödingerPoisson systemwith KohnSham potential in a bounded two- or three-dimensional domain.Nonlinear Anal. Theory Methods Appl., 41:3372, 2000.[8℄ W. Kohn. Eletroni struture of matter: Wave funtions and density fun-tionals. Rev. Mod. Phys., 71:12531266, 1999.[9℄ W. Kohn and L. J. Sham. Self-Consistent Equations Inluding Exhange andCorrelation Eets. Physial Review, 140:A1133, 1965.[10℄ E. Lieb and G. K. Pedersen. Convex multi-variable trae funtions. Rev. Math.Phys., 14:631648, 2002.[11℄ J. von Neumann. Mathematishe Grundlagen der Quantenmehanik (Math-ematial foundations of quantum mehanis), volume 38 of Grundlehren dermathematishen Wissenshaften. Springer, Berlin, 1932.[12℄ F. Nier. A Stationary Shrödinger-Poisson System Arising from the Modellingof Eletroni Devies. Forum Mathematium, 2:489510, 1990.[13℄ R. G. Parr and W. Yang. DensityFuntional Theory of Atoms and Moleules,volume 16 of International Series of Monographs on Chemistry. Oxford Uni-versity Press, Oxford, 1989.[14℄ E. Prodan and P. Nordlander. On the Kohn-Sham equations with periodibakground potentials. J. Statist. Phys., 111:967992, May 2003.[15℄ U. von Barth. Density Funtional Theory for Solids. In P. Phariseau andW. M. Temmerman, editors, The Eletroni Struture of Complex Systems,volume B 113 of NATO ASI Series. Plenum Press, New York and London,1984.Preprint 1275, Weierstrass Institute for Applied Analysis and Stohastis, Berlin 2007

https://archive.wias-berlin.de/servlets/MCRFileNodeServlet/wias_derivate_00001430/wias_preprints_1275.pdf

Monotonicity properties of the quantum mechanical particle density

Abstract

Similar works

Full text

Available Versions

Repositorium für Naturwissenschaften und Technik (TIB Hannover)

Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics