The relativistic Scott correction for atoms and molecules
Jan Philip Solovej, Thomas Østergaard Sørensen, Wolfgang L. Spitzer
aa r X i v : . [ m a t h - ph ] A ug THE RELATIVISTIC SCOTT CORRECTION FOR ATOMS ANDMOLECULES
JAN PHILIP SOLOVEJ, THOMAS ØSTERGAARD SØRENSEN, AND WOLFGANG L. SPITZER
Abstract.
We prove the first correction to the leading Thomas-Fermi energy for theground state energy of atoms and molecules in a model where the kinetic energy of theelectrons is treated relativistically. The leading Thomas-Fermi energy, established in [25],as well as the correction given here are of semi-classical nature. Our result on atoms andmolecules is proved from a general semi-classical estimate for relativistic operators withpotentials with Coulomb-like singularities. This semi-classical estimate is obtained usingthe coherent state calculus introduced in [36]. The paper contains a unified treatment ofthe relativistic as well as the non-relativistic case.
Contents
1. Introduction and main results 11.1. Main semi-classical result 52. Preliminaries 72.1. Analytic tools 72.2. Thomas-Fermi theory 113. Proof of the relativistic Scott correction for the molecular ground state energy 124. Relativistic semi-classics for potentials with Coulomb-like singularities 155. Local relativistic semi-classical estimates using new coherent states 35Appendix A. Various Proofs 41Appendix B. Estimates of semi-classical integrals 51References 581.
Introduction and main results
Our goal in this paper is to study how relativistic effects influence the energies of atomsand molecules. More specifically, we are aiming at proving a relativistic analog of the cele-brated Scott correction [29, 16, 13, 15, 30, 31, 32, 36]. At present there is no mathematicallywell-defined fully relativistically invariant theory of atoms and molecules. We will here con-sider a simplified model, which shows the relevant qualitative features of relativistic effects.In this model, these effects are introduced by treating the kinetic energy of electrons of massm by the operator √− ~ c ∆ + m c − mc instead of the standard non-relativistic Laplaceoperator − ~ ∆ / ~ is Planck’s constant. It is thesimplest of a class of models that attempts to include relativistic effects; see [12, 23]. Al-though this model does not give accurate numerical agreement with observations it is froma qualitative point of view quite realistic. Date : October 27, 2018.c (cid:13)
One of the qualitative features that our model shares with all other relativistic models isthe instability of large atoms or molecules. The natural parameter to measure relativisticcorrections in atoms and molecules is the dimensionless fine-structure constant α = e / ~ c,where e is the electron charge. As we will explain below, if Zα is too large ( Z is the atomicnumber) then atoms are unstable. In our model the critical value of Zα is too small comparedwith experimental results and with what is assumed to be the correct critical value, namely Zα = 1.Our main interest here is the behavior of the total ground state energy of large atoms andmolecules. Because of the relativistic instability problem mentioned above one cannot simplyconsider the limit of large atomic number Z . One is forced to look at the simultaneous limitof small fine-structure constant α in such a way that the product Zα remains bounded. Ofcourse, α has a fixed value which experimentally is approximately 1 / α tending to zero is strictly speaking not physically correct. Likewise, consideringthe limit of Z tending to infinity is in contradiction with the fact that the experimentallyobserved values of Z are bounded (by 92 for the stable atoms). Studying the limit Z →∞ and α → Zα bounded allows us to make a precise mathematical statementabout the asymptotics. There is numerical evidence that the asymptotics is indeed a goodapproximation to the total ground state energy for the physical values of Z and α .The first to, at least heuristically, suggest to consider Zα as a separate parameter in thelimit Z → ∞ was Schwinger [27]. In this original paper, Schwinger finds discrepancies of hisestimates of relativistic corrections with numerical evidence. Later [8], more corrections aretaken into account and excellent agreement is found. This accuracy however goes beyond arigorous mathematical treatment. We content ourselves with giving a rigorous treatment ofthe simplified model with the correct qualitative behavior.The first rigorous treatment of the limit Z → ∞ with Zα bounded was given by one ofus is the paper [25], where the leading asymptotics of the ground state energy was found. Itturns out it does not depend on Zα . The goal of the present paper is the first correction tothe leading asymptotics, i.e., the Scott correction and, in particular, to show that it dependson Zα . The work in [25] was generalized to another relativistic model in [4].We now introduce the molecular many-body Hamiltonian we consider in this paper. Lete and m denote the electric charge and mass of an electron. Let Z e = ( Z e , . . . , Z M e),where Z , . . . , Z M >
0, be the charges of the M nuclei. We consider the Born-Oppenheimerformulation where these nuclei are at fixed positions R = ( R , . . . , R M ) ∈ R M . We have N electrons. As explained above the relativistic kinetic energy of the j -th electron is equalto p − ~ c ∆ j + m c − mc , where ∆ j is the Laplacian with respect to the j -th electroncoordinate y j ∈ R , j = 1 , . . . , N . The potential energy of the electrons is composed of theattraction to the nuclei, e V ( Z e , R , y ) = M X k =1 Z k e | y − R k | , (1)and the electron-electron repulsion, X ≤ i Let us now introduce the fundamental constants. Namely, let a = ~ / me be the Bohr radius,and R ∞ = me / ~ Rydberg’s constant. Then by a change of coordinates y j → x j = y j /a ,we see that(2 R ∞ ) − H rel =: H ( Z , R ; α ) = H ( Z , . . . , Z M , R , . . . , R M ; α )= Z X j =1 hq − α − ∆ j + α − − α − − V ( Z , R , x j ) i + X ≤ i Here the error term means that |O ( Z − / ) | ≤ CZ − / , where the constant C only dependson r and M . As before, √− α − ∆ + α − − α − = − ∆ when α = 0 . Remark 2. A less detailed version of our result was announced in [35]. Remark 3. Several features of our result and its proof should be stressed:(i) The constant E TF ( z , r ) is determined in Thomas-Fermi theory.(ii) The fact that R = Z − / r is the relevant scaling of the nuclear coordinates may, aswe shall see, be understood from Thomas-Fermi theory.(iii) A characterization of the function S is given explicitly in Corollary 6 below (see alsoLemma 25). Its continuity is proved in Theorem 4.(iv) The asymptotic result is uniform in the parameters Z k α ∈ [0 , /π ], k = 1 , . . . , M .(v) The result contains, as a special case, the non-relativistic situation Z k α = 0 and, inparticular, the non-relativistic limit is controlled due to the continuity of the function S and the uniformity in the parameters Z k α . In order to get the non-relativistic limitit is important that all estimates have the correct non-relativistic behavior. This isan important issue in this work. Note that in the non-relativistic case the value S (0) = 1 / Z k α = 2 /π is understood since the function S is continuous up to the critical value 2 /π . This is, however, a less important pointsince we do not know whether the model we study gives a good description near thecritical value.The Scott correction was predicted by Scott [29] to be the first correction to theThomas-Fermi energy. In the non-relativistic setting, this was mathematically establishedby Hughes [13], Siedentop and Weikard [30, 31, 32] for atoms (and by Bach [2] for ions) andlater by Ivri˘ı and Sigal [15] for molecules. Later a different proof was given by two of us formolecules [36]. Based on methods in [15], Balodis Matesanz [3] gave a proof for the Scottcorrection of matter. The Scott correction for operators with magnetic fields was studied bySobolev [33, 34] (in the non-interacting case).In [9], Fefferman and Seco derived rigorously the second correction to Thomas-Fermi the-ory for atoms, which is of the order Z / . This was predicted by Dirac [7] and Schwinger [28].It is apparently still an open problem to prove this for molecules and to find the relativisticcorrection to this order.The main approach to proving the energy asymptotics for large atoms and molecules goesback to Lieb and Simon [21] and is to use semi-classical estimates. The Z -scaling makes itpossible to relate the many-body problem to a one-body spectral problem, which may betreated semi-classically, where the semi-classical parameter is h = Z − / . Here, severaltechniques have been developed. Lieb and Simon used Dirichlet–Neumann bracketing. Thisis however not refined enough to get beyond the leading term. The Weyl calculus [26] isthe most advanced and precise method as far as optimal semi-classical error estimates areconcerned, but it also will not directly give the Scott correction. Ivri˘ı and Sigal [15] usedFourier intergral operator techniques to establish the non-relativistic Scott correction for HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 5 molecules. Hughes and Siedentop–Weikard used methods that were designed particularlyfor spherically symmetric models, i.e., the atomic case.A simple method, which is particularly well adapted to many-body problems is that ofcoherent states. It was pioneered by Lieb [16] and Thirring [37] to give very short proofsthat Thomas-Fermi theory is correct to leading order. It is one of our major contributionshere to use an improved calculus of coherent states as developed by two of us in [36] to therelativistic setting.One feature of our work is that we give a general semi-classical estimate for relativisticone-body operators for potentials with singularities such as the Thomas-Fermi potential(see Theorem 4 below). This is derived by first proving a localised semi-classical estimatefor potentials with some smoothness (see Theorem 32). The proof here is not much differentfrom the one presented in [36] for non-relativistic Schr¨odinger operators. We do not claimthat our error estimates are sharp given the regularity we assume on the potential, but onlythat they are sufficient to prove the Scott correction. In this connection we point out thatin order to prove the Scott correction it is enough that the error relative to the leading termis smaller by more than one power of the semi-classical paramter h . In our case the relativeerror in Theorem 32 is h / .The relativistic kinetic energy is more cumbersome to work with than the Laplace operatorand large parts of the rest of our proof from [36] have to be done differently. A main issueis to be able to localise into separate regions. Since the relativistic kinetic energy is a non-local operator, localisation estimates are more involved than in the non-relativistic setting.The philosophy is that localisation errors should behave as if we were working with non-relativistic local error terms up to some exponentially small tails (see Theorem 14).The proof of the main theorem presented in Section 3 is based on the general semi-classicalestimate Theorem 4 and the use of a correlation estimate (see Theorem 17) to reduce to theone-body problem.After we had announced our results in [35], Frank, Siedentop, and Warzel [11] found aproof for the atomic case based on the method of Siedentop and Weikard [30, 31, 32], also[10] for the model studied in [4]. This approach seems to be restricted to the spherical case.This work does also not, contrary to the present work, make any special treatment of thenon-relativistic limit or the continuity of the function S .1.1. Main semi-classical result. We consider the semi-classical approximation for the relativistic operator T β ( − i h ∇ ) − V (ˆ x ) , where T β ( p ) = ( p β − p + β − − β − , β ∈ (0 , ∞ ) p , β = 0 . (3)We will consider potentials V : R → R with Coulomb singularities of the form z k | x − r k | − , k = 1 , . . . , M , at points r , . . . , r M ∈ R and with charges 0 < z , . . . , z M ≤ /π . Define d r ( x ) = min (cid:8) | x − r k | (cid:12)(cid:12) k = 1 , . . . , M (cid:9) , r = ( r , . . . , r M ) ∈ R M . (4)We assume that for some µ ≥ V satisfies (cid:12)(cid:12) ∂ η (cid:0) V ( x ) + µ (cid:1)(cid:12)(cid:12) ≤ (cid:26) C η,µ d r ( x ) − −| η | if µ = 0 C η min { d r ( x ) − , d r ( x ) − } d r ( x ) −| η | if µ = 0 (5) J. P. SOLOVEJ, T. ØSTERGAARD SØRENSEN, AND W. L. SPITZER for all x ∈ R with d r ( x ) = 0 and all multi-indices η with | η | ≤ 3, and (cid:12)(cid:12) V ( x ) − z k | x − r k | − (cid:12)(cid:12) ≤ Cr − + C (6)for | x − r k | < r min / r min = min k = ℓ | r k − r ℓ | . Note, in particular, that the Thomas-Fermipotential V TF ( z , r , · ) discussed in (35) below satisfies these requirements, by Theorem 20.So does the potential V ( x ) = π | x | − M = 1, r = 0, and d r ( x ) = | x | ).The main new result in this section is the relativistic Scott correction to the semi-classicalexpansion for potentials of this form. It will be proved in Section 4 below. The power − Theorem 4 ( Scott-corrected relativistic semi-classics). There exists a continuous,non-increasing function S : [0 , /π ] → R with S (0) = 1 / , such that for all h > , ≤ β ≤ h , T β as in (3) , and all potentials V : R → R satisfying (5) and (6) with r min > r > and max { z , . . . , z M } ≤ /π , we have (cid:12)(cid:12)(cid:12) Tr (cid:2) T β ( − i h ∇ ) − V (ˆ x ) (cid:3) − − (2 πh ) − Z (cid:2) p − V ( v ) (cid:3) − dvdp − h − M X k =1 z k S ( β / h − z k ) (cid:12)(cid:12)(cid:12) ≤ Ch − / . (7) Here, [ x ] − = min { x, } . The constant C > depends only on M , r , µ and the otherconstants in (5) and (6) .Moreover, we can find a density matrix γ , whose density ρ γ satisfies (with k · k / the L / -norm) (cid:12)(cid:12)(cid:12)(cid:12)Z ρ γ ( x ) dx − / (3 π ) − h − Z | V ( x ) − | / dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ch − / (8) and (cid:13)(cid:13) ρ γ − / (3 π ) − h − | V − | / (cid:13)(cid:13) / ≤ Ch − − / , (9) such that Tr (cid:2) ( T β ( − i h ∇ ) − V (ˆ x )) γ (cid:3) ≤ (2 πh ) − Z (cid:2) p − V ( v ) (cid:3) − dvdp + h − M X k =1 z k S ( β / h − z k )+ Ch − / . (10) Remark 5. The term proportional to h − is called the Scott correction . If β = h then itonly depends on the charges z k , k = 1 , . . . , M , of the Coulomb-singularities. Notice that thefunction in the semi-classical integral is the non-relativistic energy. This is also the reasonwhy the leading Thomas-Fermi energy is independent of β .Applying this theorem to the potential V ( x ) = π | x | − M = 1, r = 0, and d r ( x ) = | x | ), and using a simple scaling argument, gives the followingexplicit characterization of the function S in Theorem 4 (see details in Lemma 27 in Section 4below). Corollary 6 ( Characterization of the Scott-correction S ). The function S satisfies,uniformly for α ∈ [0 , /π ] , S ( α ) = lim κ → (cid:16) Tr (cid:2) H C + κ (cid:3) − − (2 π ) − Z (cid:2) p − | v | − + κ (cid:3) − dpdv (cid:17) , (11) HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 7 where H C ( α ) = ( √− α − ∆ + α − − α − − | ˆ x | − , α ∈ (0 , /π ] − ∆ − | ˆ x | − , α = 0 . (12) Remark 7. Another characterization of the function S is given in Lemma 25 in Section 4below. Remark 8. The result in Corollary 6 was proved in [24, Theorem 7.4], but only pointwise and only for α ∈ (0 , /π ). 2. Preliminaries Analytic tools. We recall here the main analytic tools which we use throughout thispaper. We do not prove all of them here but give the standard references. Various constantsare denoted by the same letter C although its value may change from one line to the next.Let p ≥ 1, then a complex-valued function f (and only those will be considered here) issaid to be in L p ( R n ) if the norm k f k p = (cid:0)R | f ( x ) | p dx (cid:1) /p is finite. We denote by h , i theinner product on L ( R n ); it is linear in the second and anti-linear in the first entry. Forany 1 ≤ p ≤ t ≤ q ≤ ∞ we have the inclusion L p ∩ L q ⊂ L t , since by H¨older’s inequality k f k t ≤ k f k λp k f k − λq with λp − + (1 − λ ) q − = t − . We denote by ˆ f the Fourier transformof f ∈ L ( R n ), given by ˆ f ( p ) = (2 π ) − n/ R e − i x · p f ( x ) dx for Schwartz functions on R n , andextended by continuity to L ( R n ).We denote x − = min { x, } , and let χ A be the characteristic function of the set A ; wewrite χ = χ ( −∞ , for the characteristic function of ( −∞ , γ a density matrix on L ( R n ) if it is a trace class operator on L ( R n ) satisfying the operator inequality ≤ γ ≤ .The density of a density matrix γ is the L -function ρ γ such that Tr( γθ ) = R ρ γ ( x ) θ ( x ) dx for all θ ∈ C ∞ ( R n ) considered as a multiplication operator.We also need an extension to many-particle states. Let ψ ∈ N N L ( R × {− , } ) be an N -body wave-function. Its one-particle density ρ ψ is defined by ρ ψ ( x ) = N X j =1 X s = ± · · · X s N = ± Z | ψ ( x , s , . . . , x N , s N ) | δ ( x j − x ) dx · · · dx N . The following two inequalities we recall are crucial in many of our estimates. They serveas replacements for the Lieb-Thirring inequality [22] used in the non-relativistic case. Theorem 9 ( Daubechies inequality). One-body case: Let m > , f ( u ) = √ u + m − m , and F ( s ) = R s [ f − ( t )] n dt , where f − denotes the inverse function of f . Assume that V ∈ L ( R n ) , and let − ∆ be the Laplacian in R n . Then Tr (cid:2)p − ∆ + m − m + V (ˆ x ) (cid:3) − ≥ − C Z F (cid:0) | V ( x ) − | (cid:1) dx , (13) where x − = min { x, } , and C is some positive constant. Many-body case: Let ψ ∈ V N L ( R ×{− , +1 } ) and let ρ ψ = ρ be its one-particle density.Then D ψ, N X j =1 (cid:2)q − ∆ j + m − m (cid:3) ψ E ≥ Z G [ ρ ( x )] dx , (14) J. P. SOLOVEJ, T. ØSTERGAARD SØRENSEN, AND W. L. SPITZER where (with C = 0 . ) G ( ρ ) = (3 / m C g [( ρ/C ) / m − ] − mρ , (15) with g ( t ) = t (1 + t ) / (1 + 2 t ) − log[ t + (1 + t ) / ] . The asymptotic behaviour of G for small, respectively large ρ is given by G ( ρ ) ∼ ρ → (3 / m ) C − / ρ / , G ( ρ ) ∼ ρ →∞ (3 / C − / ρ / . (16)By a simple scaling, and using the definition of T and (16), respectively, we see thatTr (cid:2)p − α − ∆ + m α − − mα − + V (ˆ x ) (cid:3) − (17) ≥ − Cm n/ Z | V ( x ) − | n/ dx − Cα n Z | V ( x ) − | n dx , and D ψ, N X j =1 (cid:2)q − α − ∆ j + m α − − mα − (cid:3) ψ E ≥ C Z min { m − ρ ( x ) / , α − ρ ( x ) / } dx . (18)Both (17) and (18) also holds for α = 0, where we let √− α − ∆ + m α − − mα − = − ∆ / m ,when α = 0. The original proofs of the inequalities (13) and (14) can be found in [6] (for α = 0, in [22]). Theorem 10 ( Lieb-Yau inequality). Let n = 3 . Let C > and R > and let H C,R = √− ∆ − π | ˆ x | − C/R . (19) Then, for any density matrix γ and any function θ with support in B R = { x | | x | ≤ R } wehave that Tr (cid:2) ¯ θγθH C,R (cid:3) ≥ − . C R − { / (4 πR ) Z | θ ( x ) | dx } . (20)Note that when θ = 1 on B R then the term inside the brackets {} equals 1.We will need the following new operator inequality. The proof can be found in Appendix A. Theorem 11 ( Critical Hydrogen inequality). Let n = 3 . For any s ∈ [0 , / thereexists constants A s , B s > such that √− ∆ − π | ˆ x | ≥ A s ( − ∆) s − B s . (21)We also use the following standard notation for the Coulomb energy, D ( f ) = D ( f, f ) = Z f ( x ) | x − y | − f ( y ) dxdy . Theorem 12 ( Hardy-Littlewood-Sobolev inequality). There exists a constant C suchthat D ( f ) ≤ C k f k / . (22)The sharp constant C has been found by Lieb [18]; see also [19]. It can be shown byFourier transformation that f p D ( f ) is a norm. This fact will play a role in the proofof the upper bound in our main Theorem 1.In order to localise the relativistic kinetic energy we shall use the equivalent of the IMS-formula for the operator − ∆ / m . In the sequel, as before, √− α − ∆ + m α − − mα − = − ∆ / m , when α = 0. HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 9 Theorem 13 ( Relativistic IMS formula). Let ( θ u ) u ∈M be a family of positive bounded C -functions on R with bounded derivatives, and let dµ be a positive measure on M suchthat R M θ u ( x ) dµ ( u ) = 1 for all x ∈ R . Then for any f ∈ H / ( R ) , ( f, (cid:0)p − α − ∆ + m α − − mα − (cid:1) f ) (23)= Z M ( θ u f, (cid:0)p − α − ∆ + m α − − mα − (cid:1) θ u f ) dµ ( u ) − ( f, Lf ) , where the operator L is of the form L = Z M L θ u dµ ( u ) , (24) with L θ u the bounded operator with kernel L θ u ( x, y ) = (2 π ) − m α − | x − y | − K ( mα − | x − y | ) (cid:2) θ u ( x ) − θ u ( y ) (cid:3) . (25) Here, K is a modified Bessel function of the second kind. For α = 0 , L θ u is multiplicationby ( ∇ θ u ) / m , where √− α − ∆ + m α − − mα − = − ∆ / m , when α = 0 . A proof (and the definition of K , and some of its properties) can be found in Appendix A.The following bound on the localisation error will be crucial. Theorem 14 ( Localisation error). Let Ω ⊂ R and ℓ > . Let θ be a Lipschitz continuousfunction satisfying ≤ θ ≤ , dist(Ω c , supp ∇ θ ) ≥ ℓ , and θ is constant on Ω c .Then for all m > , α ≥ there exists a positive operator Q θ such that the followingoperator inequality holds: L θ ≤ C m − k∇ θ k ∞ χ Ω + Q θ , (26) with Tr[ Q θ ] ≤ C mα − ℓ − e − mα − ℓ k∇ θ k ∞ | Ω | , (27) for a constant C > , independent of m, α, ℓ, θ , and Ω . Here, χ Ω and | Ω | are the character-istic function and the volume, respectively, of the set Ω . For α = 0 , Q θ ≡ . A proof can be found in Appendix A. Note that the first term, C m − k∇ θ k ∞ χ Ω , on theright side of (26) is similar to the error in the non-relativistic IMS formula for the operator − ∆ / m , except in this case one has k∇ θ k ∞ χ supp ∇ θ / m as the only error.When localising, we shall make use of the following. Theorem 15 ( Partition of R n ). Consider ϕ ∈ C ∞ ( R n ) with support in the unit ball {| x | ≤ } and satisfying R ϕ ( x ) dx = 1 . Assume that ℓ : R n → R is a C -map satisfying < ℓ ( u ) ≤ and k∇ ℓ k ∞ < . Let J ( x, u ) be the Jacobian of the map u x − uℓ ( u ) , i.e., J ( x, u ) = ℓ ( u ) − n (cid:12)(cid:12)(cid:12) det h ( x i − u i ) ∂ j ℓ ( u ) ℓ ( u ) + δ ij i ij (cid:12)(cid:12)(cid:12) . We set ϕ u ( x ) = ϕ (cid:0) x − uℓ ( u ) (cid:1)p J ( x, u ) ℓ ( u ) n/ . Then, for all x ∈ R n , Z R n ϕ u ( x ) ℓ ( u ) − n du = 1 , (28) and for all multi-indices η ∈ N n we have k ∂ η ϕ u k ∞ ≤ ℓ ( u ) −| η | C η max | ν |≤| η | k ∂ ν ϕ k ∞ , (29) where C η depends only on η . This is Theorem 22 in [36].We will consider potentials V : R → R with Coulomb singularities of the form z k | x − R k | − , k = 1 , . . . , M , at points R , . . . , R M ∈ R and with charges 0 < z , . . . , z M ≤ /π .Recall that (see (4); replace r by R ) d R ( x ) = min (cid:8) | x − R k | (cid:12)(cid:12) k = 1 , . . . , M (cid:9) , R = ( R , . . . , R M ) ∈ R M . (30)To treat such potentials we will need the following combination of Theorems 9 and 10. Theproof can be found in Appendix A. Theorem 16 ( Combined Daubechies-Lieb-Yau inequality). Let R , . . . , R M ∈ R ,and assume W ∈ L ( R ) satisfies W ( x ) ≥ − νd R ( x ) − Cνmα − when d R ( x ) < αm − , (31) with αν ≤ /π and m > , α ≥ , and d R as in (30) . Assume also that the minimal distancebetween nuclei satisfies min k = ℓ | R k − R ℓ | > αm − . Then Tr (cid:2)p − α − ∆ + m α − − mα − + W (ˆ x ) (cid:3) − ≥ − Cν / α / m − Cm / Z d R ( x ) >αm − | W ( x ) − | / dx − Cα Z d R ( x ) >αm − | W ( x ) − | dx , (32) where as before √− α − ∆ + m α − − mα − = − ∆ / m , when α = 0 . Finally, we come to the two inequalities which bound the many-body ground state energyin terms of a corresponding one-body energy. Theorem 17 ( Correlation inequality). Let ρ : R → R be non-negative with D ( ρ ) < ∞ and let Φ : R → R be a spherically symmetric, non-negative function with support in theunit ball such that R Φ( x ) dx < ∞ . For s > , let Φ s ( x ) = s − Φ( x/s ) . Then, for someconstant C independent of N and s , we have X ≤ i Let γ be a density matrix on L ( R ) sat-isfying γ = 2 R ρ γ ( x ) dx ≤ Z (i.e., less than or equal to the total number of electrons)with kernel ρ γ ( x ) = γ ( x, x ) . Then E ( Z , R ; α ) ≤ (cid:2)(cid:0)p − α − ∆ + α − − α − − V ( Z , R , ˆ x ) (cid:1) γ (cid:3) + D (2 ρ γ ) . (34)The factors 2 above are due to the spin degeneracy, see [17]. We denote convolution by ∗ , i.e., ( f ∗ g )( x ) = R f ( y ) g ( x − y ) dy . We also abuse notation and write ρ ∗| x | − instead of ` ρ ∗ | · | − ´ ( x ). HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 11 Thomas-Fermi theory. Consider z = ( z , . . . , z M ) ∈ R M + and r = ( r , . . . , r M ) ∈ R M . Let 0 ≤ ρ ∈ L / ( R ) ∩ L ( R ) then the (non-relativistic) Thomas-Fermi (TF) energyfunctional, E TF , is defined as E TF ( ρ ) = (3 π ) / Z ρ ( x ) / dx − Z V ( z , r , x ) ρ ( x ) dx + D ( ρ ) , where V is as in (1).By the Hardy-Littlewood-Sobolev inequality the Coulomb energy, D ( ρ ), is finite for func-tions ρ ∈ L / ( R ) ∩ L ( R ) ⊂ L / ( R ). Therefore, the TF-energy functional is well-defined.Here we only state without proof the properties about TF-theory which we use throughoutthe paper. The original proofs can be found in [21] and [16]. Theorem 19 ( Thomas-Fermi minimizer). For all z = ( z , . . . , z M ) ∈ R M + and r = ( r , . . . , r M ) ∈ R M there exists a unique non-negative ρ TF ( z , r , x ) such that R ρ TF ( z , r , x ) dx = P Mk =1 z k and E TF ( ρ TF ) = inf (cid:8) E TF ( ρ ) (cid:12)(cid:12) ≤ ρ ∈ L / ( R ) ∩ L ( R ) (cid:9) . We shall denote by E TF ( z , r ) = E TF ( ρ TF ) the TF-energy. Moreover, let V TF ( z , r , x ) = V ( z , r , x ) − ρ TF ( z , r , · ) ∗ | x | − (35) be the TF-potential, then V TF > and ρ TF > , and ρ TF is the unique solution in L / ( R ) ∩ L ( R ) to the TF-equation: V TF ( z , r , x ) = (3 π ) / ρ TF ( z , r , x ) / . (36)Very crucial for a semi-classical approach is the scaling behavior of the TF-potential. Itsays that for any positive parameter h , V TF ( z , r , x ) = h V TF ( h − z , h r , hx ) , (37) ρ TF ( z , r , x ) = h ρ TF ( h − z , h r , hx ) , (38) E TF ( z , r ) = h E TF ( h − z , h r ) . (39)By h r we mean that each coordinate is scaled by h , and likewise for h − z and hx . Bythe TF-equation (36), the equations (37) and (38) are obviously equivalent. Notice thatthe Coulomb-potential (the potential V in (1)) has the claimed scaling behavior. The restfollows from the uniqueness of the solution of the TF-energy functional.We shall now establish the crucial estimates that we need about the TF-potential. Foreach k = 1 , . . . , M we define the function W k ( z , r , x ) = V TF ( z , r , x ) − z k | x − r k | − . (40)The function W k can be continuously extended to x = r k .The first estimate in the next theorem is very similar to a corresponding estimate in [15](recall that the function d r was defined in (4)). Theorem 20 ( Estimate on V TF ). Let z = ( z , . . . , z M ) ∈ R M + and r = ( r , . . . , r M ) ∈ R M . For all multi-indices η ∈ N and all x with d r ( x ) = 0 we have (cid:12)(cid:12) ∂ ηx V TF ( z , r , x ) (cid:12)(cid:12) ≤ C η min { d r ( x ) − , d r ( x ) − } d r ( x ) −| η | , (41) where C η > is a constant which depends on η , z , . . . , z M , and M .Moreover, for | x − r k | < r min / , where r min = min k = ℓ | r k − r ℓ | , we have − C ≤ W k ( z , r , x ) ≤ Cr − + C , (42) where the constants C > here depend on z , . . . , z M , and M . Corollary 21 ( Estimate on ρ TF ∗ | x | − ∗ ( δ − Φ t ) ). Let Φ : R → R be a sphericallysymmetric, positive function with support in the unit ball and integral 1, and for t > , let Φ t ( x ) = t − Φ( x/t ) . If ρ TF ( x ) = ρ TF ( z , r , x ) then ≤ ρ TF ∗ | x | − − ρ TF ∗ | x | − ∗ Φ t ≤ (cid:26) C t min { d r ( x ) − / , d r ( x ) − } for d r ( x ) ≥ tC t / for d r ( x ) < t (43) with the function d r from (4) , and some constant C > depending on z , . . . , z M , and M . For the proof of (41) and (42) we refer to [36]. (Note that in [36] it is claimed that W k ( z , r , x ) ≥ 0. This is not correct, but the proof in [36] does give that W k ( z , r , x ) ≥ − C .)The proof of (43) can be found in Appendix A. Remark 22. As is seen from the proofs in [36] and in Appendix A, the constants inTheorem 20 and Corollary 21 only depend on z > z , . . . , z M ∈ (0 , z ].The relation of Thomas-Fermi theory to semi-classical analysis is that the semi-classicaldensity of a gas of non-interacting (non-relativistic) electrons moving in the Thomas-Fermipotential V TF is simply the Thomas-Fermi density. More precisely, the semi-classical approx-imation to the density of the projection onto the eigenspace corresponding to the negativeeigenvalues of the Hamiltonian − ∆ − V TF is2 Z p − V TF ( z , r ,x ) ≤ dp (2 π ) = 2 / (3 π ) − ( V TF ) / ( z , r , x ) = ρ TF ( z , r , x ) . (44)Here the factor two on the very left is due to the spin degeneracy. Similarly, the semi-classical approximation to the energy of the gas, i.e., to the sum of the negative eigenvaluesof − ∆ − V TF , is2 Z (cid:2) p − V TF ( z , r , x ) (cid:3) − dxdp (2 π ) = − √ π Z V TF ( z , r , x ) / dx = E TF ( z , r ) + D (cid:0) ρ TF ( z , r , · ) (cid:1) . (45)Since (by Theorem 20) the Thomas-Fermi potential V TF ( z , r , · ) in (35) satisfies (5) and (6)(uniformly for z , . . . , z M ∈ (0 , /π ]; see Remark 22), Theorem 4 implies that the densitygiven in (44) and the energy given in (45) are the leading order terms also for the relativistic gas, i.e., for the operator T β ( − i h ∇ ) − V TF , 0 ≤ β ≤ h , with T β as in (3). That the Thomas-Fermi energy is correct to leading order for T h ( − i h ∇ ) − V TF was proved in [25]. Theorem 4establishes the first correction—the Scott correction—to the leading order.3. Proof of the relativistic Scott correction for the molecular groundstate energy In this section we prove Theorem 1. Except for the correlation inequality we proceedin exactly the same manner as in the non-relativistic case [36]. In [36] correlations werecontrolled by the Lieb-Oxford inequality [20]. Applying this inequality, correlations can beestimated by the integral R ρ / involving the electronic density ρ . Using the non-relativisticLieb-Thirring inequality such an integral can be seen to be of lower order than the totalenergy. In the present relativistic case the Daubechies inequality (14) a priori only allowsus to conclude that the integral R ρ / is of the same order as the total energy. We thereforefollow a different strategy. HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 13 Proof of Theorem 1 ( Lower bound ) . Let ψ be a (normalised) ground state wave functionand let s > 0. We will use the correlation inequality (33) with ρ ( x ) = ρ TF ( Z , R , x ). Let Φ s be a function as in Theorem 17. We shall choose s = Z − / .As above (see (4)) we have d r ( x ) = min (cid:8) | x − r k | (cid:12)(cid:12) k = 1 , . . . , M (cid:9) . Note that for thephysical positions of the nuclei we then have d R ( x ) = min (cid:8) | x − R k | (cid:12)(cid:12) k = 1 , . . . , M (cid:9) = Z − / d r ( Z / x ) . From the estimate in (43) with t = Z / s we obtain from the Thomas-Fermi scaling (38)that (cid:12)(cid:12) ρ TF ( Z , R , · ) ∗ | x | − − ρ TF ( Z , R , · ) ∗ | x | − ∗ Φ s ( x ) (cid:12)(cid:12) ≤ CZ / s ( g ( x ) + Z / ) , where g ( x ) = (2 s ) − / if d R ( x ) < sd R ( x ) − / if 2 s ≤ d R ( x ) ≤ Z − / Z − / < d R ( x ) . We find from the correlation estimate (Theorem 17) that (cid:10) ψ, H ( Z , R ; α ) ψ (cid:11) ≥ Z X j =1 (cid:10) ψ, (cid:2)q − α − ∆ j + α − − α − − V ( Z , R , ˆ x j ) (cid:3) ψ (cid:11) + (cid:10) ψ, Z X j =1 (cid:0) ρ TF ( Z , R , · ) ∗ | x | − ∗ Φ s )(ˆ x j ) ψ (cid:11) − D (cid:0) ρ TF ( Z , R , · ) (cid:1) − Cs − Z ≥ (cid:2)p − α − ∆ + α − − α − − V TF ( Z , R , ˆ x ) − CZ / sg (ˆ x ) (cid:3) − − D (cid:0) ρ TF ( Z , R , · ) (cid:1) − CsZ / − Cs − Z . (46)To control the error term with g above we shall use the combined Daubechies-Lieb-Yauinequality (Theorem 16) to estimate ε Tr (cid:2)p − α − ∆ + α − − α − − V TF ( Z , R , ˆ x ) − Cε − Z / sg (ˆ x ) (cid:3) − for some 0 < ε < ε = Z − / . We use Theorem 16 with m = 1and ν = max k Z k . Then by assumption να ≤ /π . We must also check that the assumption(31) is satisfied, i.e., that for d R ( x ) < α we have − V TF ( Z , R , x ) − Cε − Z / sg ( x ) ≥ − νd R ( x ) − Cνα − . This follows from the definition of g and the estimate on the TF potential in (42) togetherwith the Thomas-Fermi scaling (37) if α < s < C ( νZ − ) ( Zα ) − ε Z, r − + 1 < C ( νZ − )( Zα ) − Z / , which, for Z large enough, is a consequence of the assumptions in the theorem and thechoices of ε and s . Note, in particular, that νZ − = max k z k ≥ M − (since P k z k = 1)and by assumption Zα ≤ min k / ( πz k ) ≤ M/π . The constants C above depend only on z , . . . , z M , and M .According to the Thomas-Fermi estimate (41), the Thomas-Fermi scaling (37), the defi-nition of g , and the choices of s and ε we have V TF ( Z , R , x ) + Cε − Z / sg ( x ) ≤ C min { d R ( x ) − , Zd R ( x ) − } . Thus the combined Daubechies-Lieb-Yau inequality gives, since ν ≤ Z and Zα ≤ M/π ,that ε Tr (cid:2)p − α − ∆ + α − − α − − V TF ( Z , R , ˆ x ) − Cε − Z / sg (ˆ x )] − ≥ − CεZ − Cε Z (cid:0) min { d R ( x ) − , Zd R ( x ) − } (cid:1) / dx − Cεα Z d R ( x ) >α (cid:0) min { d R ( x ) − , Zd R ( x ) − } (cid:1) dx ≥ − Cε ( Z + Z / ) ≥ − CεZ / . We return to the main term in (46). Using the Thomas-Fermi scaling property (37) andreplacing x by Z − / x we arrive atTr (cid:2)p − α − ∆ + α − − α − − V TF ( Z , R , ˆ x ) (cid:3) − = Z / κ − Tr (cid:2)p − β − h ∆ + β − − β − − κV TF ( z , r , ˆ x )] − , where we have chosen κ = min k πz k ≥ Zα , h = κ / Z − / , β = Z / α κ − . (47)We shall use β and h as the semi-classical parameters when we apply Theorem 4. It istherefore important that β ≤ h . This follows since β − h = ( Zα ) − κ ≥ 1. Note also that2 /π ≤ κ ≤ M/π since z k ≤ k = 1 , . . . , M , and P k z k = 1.Putting this together with the estimate above into (46) we obtain (using the inequalityTr[ X + Y ] − ≥ Tr[ X ] − + Tr[ Y ] − for operators X and Y bounded from below (with a commoncore), and the choices of ε and s ) that (cid:10) ψ, H ( Z , R ; α ) ψ (cid:11) ≥ − ε ) Z / κ − Tr (cid:2)p − β − h ∆ + β − − β − − κV TF ( z , r , ˆ x ) (cid:3) − − CεZ / − CsZ / − Cs − Z − D (cid:0) ρ TF ( Z , R , · ) (cid:1) ≥ Z / κ − Tr (cid:2)p − β − h ∆ + β − − β − − κV TF ( z , r , ˆ x ) (cid:3) − − D (cid:0) ρ TF ( Z , R , · ) (cid:1) − CZ − / . Now we apply the semi-classical result for potentials with Coulomb-like singularities fromTheorem 4 to κV TF ( z , r , · ) (recall that 2 /π ≤ κ ≤ M/π which ensures that the constantsin (5) and (6) are uniform in κ ), and the calculation in (45). Then2 Z / κ − Tr (cid:2)p − β − h ∆ + β − − β − − κV TF ( z , r , ˆ x ) (cid:3) − = Z / (cid:16) E TF ( z , r ) + D (cid:0) ρ TF ( z , r , · ) (cid:1)(cid:17) + 2 M X k =1 Z k S ( Z k α ) + O ( Z − / )= E TF ( Z , R ) + D (cid:0) ρ TF ( Z , R , · ) (cid:1) + 2 M X k =1 Z k S ( Z k α ) + O ( Z − / ) . Note here that κ cancels in the leading semi-classical term and in the Scott-term (the termwith S ). Also, 2 /π ≤ κ ≤ M/π ensures that the error is uniform in κ . Here we have againused the TF scaling E TF ( Z , R ) = Z / E TF ( z , r ) and D (cid:0) ρ TF ( Z , R , · ) (cid:1) = Z / D (cid:0) ρ TF ( z , r , · ) (cid:1) .This finishes the proof of the lower bound. (cid:3) HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 15 Proof of Theorem 1 ( Upper bound ) . The starting point now is Lieb’s variational princi-ple, Theorem 18. By a simple rescaling the variational principle states that for any densitymatrix γ on L ( R ) with 2 Tr γ ≤ Z we have E ( Z , R ; α ) ≤ Z / Tr (cid:2)(cid:0)p − α − Z − ∆ + α − Z − / − α − Z − / − V ( z , r , ˆ x ) (cid:1) γ (cid:3) + Z / D (2 Z − ρ γ ) . As for the lower bound we bring the TF-potential into play: Z − / E ( Z , R ; α ) ≤ (cid:2)(cid:0)p − α − Z − ∆ + α − Z − / − α − Z − / − V TF ( z , r , ˆ x ) (cid:1) γ (cid:3) + ZD (cid:0) Z − ρ γ − ρ TF ( z , r , · ) (cid:1) − ZD (cid:0) ρ TF ( z , r , · ) (cid:1) = 2 κ − Tr (cid:2)(cid:0)p − β − h ∆ + β − − β − − κV TF ( z , r , ˆ x ) (cid:1) γ (cid:3) + ZD (cid:0) Z − ρ γ − ρ TF ( z , r , · ) (cid:1) − ZD (cid:0) ρ TF ( z , r , · ) (cid:1) , (48)where κ , h , and β are chosen as in (47) in the proof of the lower bound. Note that with thischoice of h and κ we have from (36) that2 / (3 π h ) − ( κV TF ( z , r , x )) / = Zρ TF ( z , r , x ) / . We now choose a density matrix e γ according to Theorem 4 with V ( x ) = κV TF ( z , r , x ).Since R ρ TF ( z , r , x ) dx = P Mk =1 z k = 1 we see from (8) that 2 Tr e γ ≤ Z (1 + CZ − / − / )(recall that κ − ≤ π/ γ = (1 + CZ − / − / ) − e γ we see that thecondition 2 Tr γ ≤ Z is satisfied.Using the Hardy-Littlewood-Sobolev and (9) inequalities we find that ZD (cid:0) Z − ρ e γ − ρ TF ( z , r , · ) (cid:1) ≤ CZ − (cid:13)(cid:13) ρ e γ − Zρ TF ( z , r , · ) / (cid:13)(cid:13) / ≤ CZ / − / , and thus ZD (cid:0) Z − ρ γ − ρ TF ( z , r , · ) (cid:1) ≤ C (1 + CZ − / − / ) − ZD (cid:0) Z − ρ e γ − ρ TF ( z , r , · ) (cid:1) + CZ / − / D (cid:0) ρ TF ( z , r , · ) (cid:1) ≤ CZ / − / , (49)where we have used the triangle inequality for √ D , and that D (cid:0) ρ TF ( z , r , · ) (cid:1) ≤ C .Finally, if we use (10) and (45) we get as for the lower bound that2 Z / κ − Tr (cid:2)(cid:0)p − β − h ∆ + β − − β − − κV TF ( z , r , ˆ x ) (cid:1)e γ (cid:3) ≤ E TF ( Z , R ) + D (cid:0) ρ TF ( Z , R , · ) (cid:1) + 2 M X j =1 Z k S ( Z k α ) + O ( Z − / ) . Since E TF ( Z , R ) ≥ − CZ / and D (cid:0) ρ TF ( Z , R , · ) (cid:1) ≥ e γ replaced by γ . This together with D (cid:0) ρ TF ( Z , R , · ) (cid:1) = Z / D (cid:0) ρ TF ( z , r , · ) (cid:1) , (48), and(49) finishes the proof of the upper bound. (cid:3) The function S is continuous and non-increasing, and S (0) = 1 / 4, according to Theorem 4.This finishes the proof of Theorem 1.4. Relativistic semi-classics for potentials with Coulomb-like singularities In this section we prove Theorem 4. The theorem will follow from using Theorem 23 below(a rescaled version of the local semi-classical results for regular potentials in Theorem 32 inSection 5 below). We localise (Theorem 13) the operator using multi-scale analysis (The-orem 15), and control the localisation errors (Theorem 16). Near the singularities of the potential, we compare with the Coulomb potential. To be able to do this, we first prove aScott-corrected semi-classical result for a localised relativistic Hydrogen operator (Lemma 25below). The ingredients of the proof of the latter are the same (rescaled semi-classics, local-isation and multi-scale analysis, and estimating localisation errors). Theorem 23 ( Rescaled semi-classics). Let n ≥ and let φ ∈ C n +40 ( R n ) be supported in aball B ℓ of radius ℓ > . Let V ∈ C ( B ℓ ) be a real potential, and let T β ( p ) = p β − p + β − − β − be the kinetic energy. Let H β = T β ( − i h ∇ ) + V (ˆ x ) , h > , and σ β ( v, q ) = T β ( q ) + V ( v ) .Then for all h, β, f > with βf ≤ , we have (cid:12)(cid:12)(cid:12)(cid:12) Tr[ φH β φ ] − − (2 πh ) − n Z φ ( v ) σ β ( v, q ) − dvdq (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ch − n +6 / f n +4 / ℓ n − / , (50) where the constant C is independent of β and depends only on sup | η |≤ n +4 k ℓ | η | ∂ η φ k ∞ and sup | η |≤ k f − ℓ | η | ∂ η V k ∞ . (51) Moreover, there exists a density matrix γ such that Tr[ γφH β φ ] ≤ (2 πh ) − n Z φ ( v ) σ β ( v, q ) − dvdq + Ch − n +6 / f n +4 / ℓ n − / . (52) The density ρ γ satisfies (cid:12)(cid:12)(cid:12) ρ γ ( x ) − (2 πh ) − n ω n | V − | n/ (2 + β | V − | ) n/ ( x ) (cid:12)(cid:12)(cid:12) ≤ Ch − n +9 / f n − / ℓ − / (53) for (almost) all x ∈ B ℓ , and (cid:12)(cid:12)(cid:12)(cid:12)Z φ ( x ) ρ γ ( x ) dx − (2 πh ) − n ω n Z φ ( x ) | V − | n/ (2 + β | V − | ) n/ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ch − n +6 / f n − / ℓ n − / , (54) where ω n is the volume of the unit ball B in R n . The constants C > in the above estimatesagain depend on the parameters as in (51) .Proof. We introduce the unitary scaling operator ( U ψ )( x ) = ℓ − n/ ψ ( ℓ − x ). Then U ∗ φ (cid:2) T β ( − i h ∇ ) + V (ˆ x ) (cid:3) φ U = f φ ℓ (cid:2) T βf ( − i hf − ℓ − ∇ ) + V f,ℓ (ˆ x ) (cid:3) φ ℓ , where φ ℓ ( x ) = φ ( ℓx ), and V f,ℓ ( x ) = f − V ( ℓx ). Thus,Tr[ φH β φ ] − = f Tr h φ ℓ (cid:2) T βf ( − i hf − ℓ − ∇ ) + V f,ℓ (ˆ x ) (cid:3) φ ℓ i − . Note that φ ℓ and V f,ℓ are supported in a ball of radius 1 and that for all multi-indices η , k ∂ η φ ℓ k ∞ = k ℓ | η | ∂ η φ k ∞ and k ∂ η V f,ℓ k ∞ = f − k ℓ | η | ∂ η V k ∞ . Let σ f,ℓ,β ( u, q ) = T βf ( q ) + V f,ℓ ( u ). By Theorem 32 in Section 5 below there is a constant C depending on the parameters as in (51) so that, as long as βf ≤ (cid:12)(cid:12)(cid:12)(cid:12) Tr[ φH β φ ] − − (2 πhf − ℓ − ) − n f Z φ ℓ ( u ) σ f,ℓ,β ( u, q ) − dudq (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cf ( hf − ℓ − ) − n +6 / . (55)A simple change of variables gives(2 πhf − ℓ − ) − n f Z φ ℓ ( u ) σ f,ℓ,β ( u, q ) − dudq = (2 πh ) − n Z φ ( v ) σ β ( v, q ) − dvdq , and we have proved (50). HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 17 Now, let γ f,ℓ,β be the density matrix for φ ℓ (cid:2) T βf ( − i hf − ℓ − ∇ ) + V f,ℓ (ˆ x ) (cid:3) φ ℓ , which ac-cording to Lemma 34 satisfies f Tr h φ ℓ (cid:2) T βf ( − i hf − ℓ − ∇ ) + V f,ℓ (ˆ x ) (cid:3) φ ℓ γ f,ℓ,β i ≤ (2 πhf − ℓ − ) − n f Z φ ℓ ( u ) σ f,ℓ,β ( u, q ) − dudq + C ( hf − ℓ − ) − n +6 / , (cid:12)(cid:12)(cid:12) ρ γ f,ℓ,β ( x ) − (2 πhf − ℓ − ) − n ω n | V f,ℓ ( x ) − | n/ (2 + βf | V f,ℓ ( x ) − | ) n/ ( x ) (cid:12)(cid:12)(cid:12) ≤ C ( hf − ℓ − ) − n +9 / , (cid:12)(cid:12)(cid:12)(cid:12)Z φ ℓ ( x ) ρ γ f,ℓ,β ( x ) dx − (2 πhf − ℓ − ) − n ω n Z φ ℓ ( x ) | V f,ℓ ( x ) − | n/ (2 + βf | V f,ℓ ( x ) − | ) n/ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( hf − ℓ − ) − n +6 / . The density matrix γ = U γ f,ℓ,β U ∗ , whose density is ρ γ ( x ) = ℓ − n ρ γ f,ℓ,β ( x/ℓ ), then satisfiesthe properties in (52)–(54). (cid:3) Multi-scale Analysis. The rescaled semi-classics of Theorem 23 will be used in balls ofvarying radius. This idea goes back to Ivri˘ı [15, 14]. We introduce a variable scale ℓ and acorresponding family of localisation functions { ϕ u } u ∈ R , which will also be used in the proofof Theorem 4. Definition 24 ( Scale for multi-scale analysis). Let < ℓ < be a parameter that weshall choose explicitly below, and let r , . . . , r M ∈ R . Define ℓ ( x ) = (cid:16) M X k =1 ( | x − r k | + ℓ ) − / (cid:17) − . (56)Note that ℓ is a smooth function (due to ℓ ) with0 < ℓ ( x ) < / k∇ ℓ k ∞ < / . (57)Note also that in terms of the function d ≡ d r from (4) we have (1 + M ) − ℓ ≤ (1 + M ( d ( x ) + ℓ ) − / ) − ≤ ℓ ( x ) ≤ ( d ( x ) + ℓ ) / . (58)In particular, we have ℓ ( x ) ≥ (1 + M ) − min { d ( x ) , } . (59)We fix a localisation function ϕ ∈ C ∞ ( R ) with support in {| x | < } and such that R ϕ ( x ) dx = 1. According to Theorem 15 we can find a corresponding family of func-tions ϕ u ∈ C ∞ ( R ), u ∈ R , where ϕ u is supported in the ball {| x − u | < ℓ ( u ) } , with theproperties that Z R ϕ u ( x ) ℓ ( u ) − du = 1 and k ∂ η ϕ u k ∞ ≤ Cℓ ( u ) −| η | , (60)for all multi-indices η , where C > η and ϕ . For d ( u ) > ℓ we have ℓ ( u ) ≤√ d ( u ) / x with | x − u | < ℓ ( u ) we have (note that d ( u ) ≤ d ( x ) + | x − u | and √ / < 1) that ℓ ( u ) < d ( u ) and d ( u ) ≤ Cd ( x ) . (61)As a first step towards the Scott correction for Coulomb-type potentials we deal withthe Scott correction for a single relativistic Hydrogen atom. This method for proving theexistence of a Scott correction in the semi-classical expansion of the sum of eigenvalues of an operator with a (homogeneous) singular potential without explicitly knowing theeigenvalues was first used by Sobolev [33] when studying (non-relativistic) operators withmagnetic fields. Lemma 25 ( Scott-corrected localised Hydrogen). There exists a non-increasing func-tion S : [0 , /π ] → R , with S (0) = 1 / , such that, if φ r ( x ) = φ ( x/r ) , r ∈ (0 , ∞ ) , with φ ∈ C ( R ) , ≤ φ ≤ , satisfying √ − φ ∈ C ( R ) and φ ( x ) = (cid:26) for | x | ≤ for | x | ≥ , then there exists C > depending only on φ such that for all α ∈ [0 , /π ] and r ∈ (0 , ∞ ) , (cid:12)(cid:12)(cid:12)(cid:12) Tr[ φ r H C ( α ) φ r ] − − (2 π ) − Z φ r ( v ) (cid:2) q − | v | − (cid:3) − dvdq − S ( α ) (cid:12)(cid:12)(cid:12)(cid:12) < Cr − / , (62) where H C ( α ) = ( √− α − ∆ + α − − α − − | ˆ x | − , α ∈ (0 , /π ] − ∆ − | ˆ x | − , α = 0 . (63)As emphasised in Remark 5, the function in the semi-classical integral in (62) is the non-relativistic energy. See also Lemma 27 below for an alternative description of the function S . Remark 26. A result similar to the one in Lemma 25 was proved in [24, Theorem 7.1], butwithout uniform control in α and only for α ∈ (0 , /π ). Proof of Lemma 25. We fix α ∈ [0 , /π ] and write H C = H C ( α ). We define for r > S r = Tr (cid:2) φ r H C ( α ) φ r (cid:3) − − (2 π ) − Z φ r ( v ) (cid:2) q − | v | − (cid:3) − dvdq . (64)We will show that S r has a limit as r → ∞ .Let R > r . We estimate the difference between Tr[ φ R H C φ R ] − and Tr[ φ r H C φ r ] − semi-classically. The leading semi-classical term involves the relativistic energy which is thencompared to the non-relativistic energy. Below all constants will depend only on φ and inparticular not on α ∈ [0 , /π ].Denote ψ r = p − φ r . By the relativistic IMS formula (23), H C = φ r H C φ r + ψ r H C ψ r − L φ r − L ψ r , where L φ r and L ψ r are given by (24) and (25) ( M = { , } ). We multiply with φ R and getthat φ R H C φ R = φ r H C φ r + φ R ψ r H C ψ r φ R − φ R ( L φ r + L ψ r ) φ R . We have used that φ R φ r = φ r since R > r . Now, let γ R = χ ( φ R H C φ R ) be the projectiononto the negative part of φ R H C φ R . Then, by the variational principle and Theorem 14 (with m = 1, ℓ = r , Ω = B (0 , r ), and θ = φ r and ψ r , respectively),Tr[ φ R H C φ R ] − = Tr[ γ R φ r H C φ r ] + Tr[ γ R φ R ψ r H C ψ r φ R ] − Tr[ γ R φ R ( L φ r + L ψ r ) φ R ] ≥ Tr[ γ R φ r ( H C − Cr − ) φ r ] + Tr[ γ R φ R ψ r ( H C − Cr − φ r ) ψ r φ R ] (65) − Cr − . HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 19 Here, C is independent of α . We treat the part of the localisation error coming from thefirst term in (65). We split H C = (1 − ε ) H C + εH C for some 0 < ε < M = 1, R = 0, d ( x ) = | x | , m = 1 and ν = 1),Tr[ γ R φ r ( εH C − Cr − ) φ r ]= ε Tr (cid:2) ( φ r γ R φ r ) (cid:8)p − α − ∆ + α − − α − − ( | ˆ x | − + Cr − ε − ) (cid:9)(cid:3) ≥ − Cε (cid:16) α / + Z α< | x | < r ( | x | − + Cε − r − ) / dx + α Z α< | x | < r ( | x | − + Cε − r − ) dx (cid:17) ≥ − Cε (cid:16) r / + ε − / r − + ε − r − ) , assuming ε − r − ≤ Cα − and using that α ≤ /π . We may choose ε = r − if we assumethat r > ε − r − = r − < ≤ α − /π ). We then obtainTr[ γ R φ r ( εH C − Cr − ) φ r ] ≥ − Cr − / . As a result, we have shown thatTr[ φ R H C φ R ] − ≥ (1 − ε ) Tr[ γ R φ r H C φ r ] + Tr[ γ R φ R ψ r ( H C − Cr − φ r ) ψ r φ R ] − Cr − / ≥ Tr[ φ r H C φ r ] − + Tr[ φ R ψ r ( H C − Cr − φ r ) ψ r φ R ] − − Cr − / . We will treat the term Tr[ φ R ψ r ( H C − Cr − φ r ) ψ r φ R ] − by our semi-classical estimatesin Section 5 below. We first rescale. Define the unitary scaling operator ( U ϕ )( x ) = R − / ϕ ( R − x ). Then e H C : = U ∗ ( H C − Cr − φ r ) U = R − (cid:0)p − α − ∆ + R α − − Rα − − | ˆ x | − − CRr − φ r/R (ˆ x ) (cid:1) = R − (cid:0) T β ( − i h ∇ ) − | ˆ x | − − CRr − φ r/R (ˆ x ) (cid:1) (66)with β = α R − ( < R − ) and h = R − / . Let φ R,r = φ R ψ r = φ R p − φ r and ψ ( x ) = φ R,r ( Rx ) (see (3) for T β ). In this way, φ R ψ r ( H C − Cr − φ r ) ψ r φ R and ψ e H C ψ are unitarilyequivalent.Now, let ℓ and ϕ u be the functions in (56) and (60), respectively, when M = 1, r = 0,and ℓ = h = R − . By another relativistic IMS localisation we get that ψ e H C ψ = R − Z r/ R< | u | < / ψϕ u (cid:0) T β ( − i h ∇ ) − | ˆ x | − − CRr − φ r/R (ˆ x ) (cid:1) ϕ u ψ ℓ ( u ) − du − R − Z r/ R< | u | < / ψL ϕ u ψ ℓ ( u ) − du . We have used that ψϕ u = 0 for | u | 6∈ [ r/ R, / ℓ ( u ) ≤ ( | u | + ℓ ) / (see (58)) andsupp ψ ⊂ { r/R ≤ | x | ≤ } , supp ϕ u ⊂ {| x − u | ≤ ℓ ( u ) } .Concerning L ϕ u , Theorem 14 with ℓ = ℓ ( u ) / m = R , and Ω = Ω u = {| x − u | ≤ ℓ ( u ) / } gives L ϕ u ≤ CR − ℓ ( u ) − χ Ω u + Q ϕ u , with Tr[ Q ϕ u ] ≤ CRα − ℓ ( u ) − e − α − Rℓ ( u ) / . (67)Here we have used (60).Notice that if the supports of ϕ u and ϕ u ′ overlap then | u − u ′ | ≤ ℓ ( u ) + ℓ ( u ′ ) and thus ℓ ( u ′ ) ≤ ℓ ( u ) + k∇ ℓ k ∞ ( ℓ ( u ) + ℓ ( u ′ )) . (68)Therefore, since k∇ ℓ k ∞ < / 2, we have that ℓ ( u ′ ) ≤ Cℓ ( u ) and thus ℓ ( u ) − ≤ Cℓ ( u ′ ) − .Similarly, ℓ ( u ) ≤ Cℓ ( u ′ ), and so χ Ω u ≤ χ {| x − u |≤ Cℓ ( u ′ ) } if the supports of ϕ u and ϕ u ′ overlap.Using this and (60) we get for all x ∈ R Z (cid:0) ℓ ( u ) − χ Ω u ( x ) (cid:1) ℓ ( u ) − du = Z (cid:0) ℓ ( u ) − χ Ω u ( x ) (cid:1) (cid:0) Z ϕ u ′ ( x ) ℓ ( u ′ ) − du ′ (cid:1) ℓ ( u ) − du ≤ C Z ϕ u ′ ( x ) ℓ ( u ′ ) − ϕ u ′ ( x ) ℓ ( u ′ ) − du ′ . (69)Rewriting the last integral with u as integration variable we get ψ e H C ψ ≥ R − Z ψϕ u (cid:0) T β ( − i h ∇ ) − | ˆ x | − − Ch ℓ ( u ) − (cid:1) ϕ u ψ ℓ ( u ) − du − R − Z ψQ ϕ u ψ ℓ ( u ) − du. Here we have also used that Rr − φ r/R ( x ) ≤ Ch ℓ ( u ) − for x in the support of ϕ u . Thisis a consequence of ℓ ( u ) ≤ | u | + ℓ ≤ | x | + ℓ ( u ) + ℓ for x in the support of ϕ u whichimplies that ℓ ( u ) ≤ | x | + ℓ ≤ Cr/R for x in the support of ϕ u and φ r/R .We will now use Theorem 23 (with φ = ψϕ u , ℓ = ℓ ( u ), B ℓ = {| x − u | ≤ ℓ ( u ) } , f = f ( u ) = | u | − / ) on ψϕ u (cid:0) T β ( − i h ∇ ) − | ˆ x | − − Ch ℓ ( u ) − (cid:1) ϕ u ψ , for u fixed with | u | ∈ [ r/ R, / k ∂ ηx ( ψϕ u ) k ∞ ≤ C η ℓ ( u ) −| η | for all η ∈ N . (70)This follows from (60), (61), and the estimate | ∂ η ψ ( x ) | ≤ C η | x | −| η | . It suffices to checkthe latter for 1 ≤ | x | ≤ r/R ≤ | x | ≤ r/R , due to the support properties of ψ .Furthermore, for r > | x | − + Ch ℓ ( u ) − is smooth (as a function of x ) on B ℓ (use (58), ℓ = R − , and | u | ≥ r/ R ), and satisfiessup | x − u | <ℓ ( u ) (cid:12)(cid:12) ∂ ηx ( | x | − + Ch ℓ ( u ) − ) (cid:12)(cid:12) ≤ C η f ( u ) ℓ ( u ) −| η | for all η ∈ N , (71)with f ( u ) = | u | − / . For the Coulomb potential, this is trivial. For the other term, only thestatement for η = 0 is non-trivial; it follows from (59), h = R − / , and | u | ≥ r/ R . Finally,the condition f ( u ) β ≤ r ≥ | u | ≥ r/ R and β < R − . HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 21 From Theorem 23 we conclude thatTr[ φ R ψ r ( H C − Cr − φ r ) ψ r φ R ] − = Tr[ ψ e H C ψ ] − ≥ R − (2 πh ) − Z r/ R< | u | < / ψ ( v ) ϕ u ( v ) (cid:2) T β ( q ) − | v | − − Ch ℓ ( u ) − ] − ℓ ( u ) − dudvdq − CR − h − / Z r/ R< | u | < / f ( u ) − / ℓ ( u ) − − / du − R − Z r/ R< | u | < / Tr (cid:2) ψQ ϕ u ψ (cid:3) ℓ ( u ) − du . Integrating the semi-classical error using f ( u ) = | u | − / , (59), and R > r gives the lowerbound − CR − h − / ( R/r ) / = − Cr − / .From (67) it follows, using (59), α ≤ /π , and R > r , that R − Z r/ R< | u | < / Tr (cid:2) ψQ ϕ u ψ (cid:3) ℓ ( u ) − du ≤ C Z r/ R< | u | < / α − e − α − Rℓ ( u ) / ℓ ( u ) − du ≤ Cr − e − r/ . Since supp ϕ u ⊂ { v | | u − v | ≤ ℓ ( u ) } and | u | ≤ / | v | ≤ | u | + ℓ ( u ) ≤ Cℓ ( u ) onsupp ϕ u . Using this, integrating in u (using (60)), we get R − (2 πh ) − Z r/ R< | u | < / ψ ( v ) ϕ u ( v ) (cid:2) T β ( q ) − | v | − − Ch ℓ ( u ) − ] − ℓ ( u ) − dudvdq ≥ R − πh ) Z ψ ( v ) (cid:2)p β − q + β − − β − − | v | − − Ch | v | − ] − dvdq . In order to compare this latter relativistic semi-classical expression with the non-relativistic semi-classical one we use the inequality | x − − y − | ≤ | x − y | and a Taylor expansionof √ t + 1 − Z (cid:12)(cid:12)(cid:12)(cid:2) q − a (cid:3) − − [ p β − q + β − − β − − a − b ] − (cid:12)(cid:12)(cid:12) dq ≤ Cβ ( β ( a + b ) + 2( a + b )) / + Cb ( β ( a + b ) + 2( a + b )) / (72)for all a, b > 0. This gives, using h = R − and β < R − , that (cid:12)(cid:12)(cid:12)(cid:12)Z ψ ( v ) (cid:16)(cid:2) q − | v | − (cid:3) − − (cid:2)p β − q + β − − β − − | v | − − Ch | v | − (cid:3) − (cid:17) dvdq (cid:12)(cid:12)(cid:12)(cid:12) ≤ CR − Z r/R< | v | < | v | − / dv ≤ C ( Rr ) − / , (73)since R > r ≥ R − (2 πh ) − Z r/ R< | u | < / ψ ( v ) ϕ u ( v ) (cid:2) T β ( q ) − | v | − − Ch ℓ ( u ) − ] − ℓ ( u ) − dudvdq ≥ (2 π ) − Z φ R,r ( v ) (cid:2) q − | v | − ] − dvdq − Cr − / . Summarizing, we have proved that there exists a constant C = C ( φ ), independent of α ∈ [0 , /π ], such that for r large enough, and R > r ,Tr[ φ R H C φ R ] − ≥ Tr[ φ r H C φ r ] − + (2 π ) − Z φ R,r ( v ) (cid:2) q − | v | − ] − dvdq − Cr − / . (74)Next, we want to bound Tr[ φ R H C φ R ] − from above by Tr[ φ r H C φ r ] − by constructing adensity matrix. To this end, we first set γ r = χ ( φ r H C φ r ). Then we let e γ u be the den-sity matrix we get when we use Theorem 23 for the rescaled operator ψϕ u e H C ϕ u ψ (nowwith e H C = U ∗ H C U with U as in (66)), for fixed u with | u | ∈ [ r/ R, / γ u = U ϕ u e γ u ϕ u U ∗ . Finally, we define γ = φ r γ r φ r + Z r/ R< | u | < / ψ r γ u ψ r ℓ ( u ) − du . (75)Since ≤ e γ ≤ and R ϕ u ( x ) ℓ ( u ) − du = 1, ≤ Z γ u ℓ ( u ) − du ≤ , and so we see, by multiplication with ψ r on both sides, that ≤ γ ≤ . Also, γ is triviallytrace class. By the variational principle we obtain thatTr[ φ R H C φ R ] − ≤ Tr[ φ R H C φ R γ ]= Tr (cid:2) φ R φ r H C φ r φ R χ ( φ r H C φ r ) (cid:3) + Z r/ R< | u | < / Tr[ ψ r φ R H C φ R ψ r γ u ] ℓ ( u ) − du ≤ Tr[ φ r H C φ r ] − + Z r/ R< | u | < / Tr[ ψϕ u e H C ϕ u ψ e γ u ] ℓ ( u ) − du . Here we have used that φ R φ r H C φ r φ R = φ r H C φ r , since R > r , and again scaled theoperators inside the trace in the integrand. Using Theorem 23 we can bound the integralfrom above by R − (2 πh ) − Z ψ ( v ) ϕ u ( v ) (cid:2) T β ( q ) − | v | − (cid:3) − ℓ ( u ) − dudvdq + CR − h − / Z r/ R< | u | < / f ( u ) − / ℓ ( u ) − − / du . As in the case of the lower bound, the error term is bounded by Cr − / .Integrating with respect to u in the semi-classical expression above, changing back coor-dinates, and using (73), we conclude thatTr[ φ R H C φ R ] − ≤ Tr[ φ r H C φ r ] − + (2 π ) − Z φ R,r ( v ) (cid:2) q − | v | − (cid:3) − dvdq + Cr − / . (76) HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 23 Combining (74) and (76) we have shown that for R > r , (cid:12)(cid:12) S R − S r (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) Tr[ φ R H C φ R ] − − Tr[ φ r H C φ r ] − + (2 π ) − Z (cid:0) φ r ( v ) − φ R ( v ) (cid:1) (cid:2) q − | v | − (cid:3) − dvdq (cid:12)(cid:12)(cid:12) ≤ Cr − / . Hence, { S n } n ∈ N is a Cauchy-sequence and with S = S ( α ) the limiting value we have (cid:12)(cid:12) S r − S (cid:12)(cid:12) ≤ Cr − / . This proves (62). That S is non-increasing follows from the fact that T α ( p ) (see (3)) isdecreasing in α . Finally, that S (0) = 1 / (cid:3) Proof of Theorem 4. Using the combined Daubechies-Lieb-Yau inequality (see Theorem 16)with α = β / h − ( ≤ 1) and m = h − we may assume that h is bounded by some constant,which we may choose small depending on M and r , using that z k ≤ /π , k = 1 , . . . , M ,and that S is a bounded function (since it is non-increasing; see Lemma 25).In order to control the region close to and far away from all the nuclei we introducelocalisation functions θ ± ∈ C ( R ) with the properties that 0 ≤ θ ± ≤ θ − + θ = 1,(2) θ − ( t ) = 1 if t < θ − ( t ) = 0 for t > < r < r / < r < R and define Φ ± ( x ) = θ ± ( d ( x ) /R ) and φ ± ( x ) = θ ± ( d ( x ) /r )(with d = d r as in (4)). We choose (assuming h is small enough) r = δ − h and R = (cid:26) Ch − , if µ = 0 R µ , if µ = 0 , (77)where δ = h < / R µ = Cµ − is chosen such that − V ( x ) ≥ d ( x ) ≥ R µ (see(5)). We will keep writing δ and R in the calculations below to show why these choices areoptimal. Clearly, Φ − + Φ = 1, φ − + φ = 1, and φ − + Φ − φ + Φ = 1. Note also that φ − ( x ) = M X k =1 θ r,k ( x ) with θ r,k ( x ) = θ − ( | x − r k | /r ) . Step 1: Lower bound on the quantum energy. By the relativistic IMS formula (23) and Theorem 14 with m = h − , α = β / h − ( ≤ ℓ = R , Ω = { d ( x ) ≤ R } , and θ = Φ ± respectively, or ℓ = r , Ω = { d ( x ) ≤ r } ,and θ = θ r,k , k = 1 , . . . , M , or θ = φ + respectively, we find that T β ( − i h ∇ ) − V (ˆ x )= Φ + (cid:0) T β ( − i h ∇ ) − V (ˆ x ) (cid:1) Φ + + Φ − (cid:0) T β ( − i h ∇ ) − V (ˆ x ) (cid:1) Φ − − L Φ − − L Φ + = M X k =1 θ r,k (cid:0) T β ( − i h ∇ ) − V (ˆ x ) (cid:1) θ r,k + Φ − φ + (cid:0) T β ( − i h ∇ ) − V (ˆ x ) (cid:1) φ + Φ − + Φ + (cid:0) T β ( − i h ∇ ) − V (ˆ x ) (cid:1) Φ + − Φ − ( M X k =1 L θ r,k + L φ + )Φ − − L Φ − − L Φ + , (78)with L Φ ± ≤ Ch k∇ Φ ± k ∞ χ { d ( x ) ≤ R } + Q Φ ± , (79)Tr[ Q Φ ± ] ≤ Cβ − R − e − ( β / h ) − R k∇ Φ ± k ∞ (cid:12)(cid:12) { d ( x ) ≤ R } (cid:12)(cid:12) , (80) and (with, by abuse of notation, L φ − = P Mk =1 L θ r,k ) L φ ± ≤ Ch k∇ φ ± k ∞ χ { d ( x ) ≤ r } + Q φ ± , (81)Tr[ Q φ ± ] ≤ Cβ − r − e − ( β / h ) − r k∇ φ ± k ∞ (cid:12)(cid:12) { d ( x ) ≤ r } (cid:12)(cid:12) . (82)Using (cid:12)(cid:12) { d ( x ) ≤ R } (cid:12)(cid:12) ≤ πM R , k∇ Φ ± k ∞ ≤ CR − (and the corresponding estimates for r and φ ± ), β ≤ h , and h small, it follows thatTr[ Q Φ ± ] ≤ Ch R − e − h − R/ ≤ C N h N , Tr[ Q φ ± ] ≤ Ch r − e − h − r/ ≤ C N h N , for any N > (cid:2) T β ( − i h ∇ ) − V (ˆ x ) (cid:3) − ≥ M X k =1 Tr (cid:2) θ r,k (cid:0) T β ( − i h ∇ ) − V (ˆ x ) − Ch r − (cid:1) θ r,k (cid:3) − + Tr (cid:2) Φ − φ + (cid:0) T β ( − i h ∇ ) − V (ˆ x ) − Ch r − χ { d ( x ) ≤ r } − Ch R − (cid:1) φ + Φ − (cid:3) − + Tr (cid:2) Φ + (cid:0) T β ( − i h ∇ ) − V (ˆ x ) − Ch R − χ { d ( x ) ≤ R } (cid:1) Φ + (cid:3) − − Ch − / . (83)Each of the first three terms above will be compared to the corresponding semi-classicalexpression. We shall treat the three terms by different methods. The first term will becalculated using the Scott correction for Hydrogen in Lemma 25. The second term will becomputed using the local rescaled semi-classics in Theorem 23. The last term is an errorterm which we will treat first. Control of the third term in (83) . We use the Daubechies inequality (17) with m = h − and α = β / h − ( ≤ µ = 0 we obtain, using the choice (77) of R ,Tr (cid:2) Φ + ( T β ( − i h ∇ ) − V (ˆ x ) − Ch R − χ { d ( x ) ≤ R } )Φ + (cid:3) − ≥ − Ch − M Z | x | >R | x | − / dx − CM Z | x | >R | x | − dx − Ch R − − Ch R − ≥ − C (cid:0) h − R − / + R − + h R − − h R − (cid:1) ≥ − Ch / . (84)The case µ = 0 gives a smaller error since − V ≥ + in this case. Control of the first term in (83) . Using (6) and (77) we have M X k =1 Tr (cid:2) θ r,k (cid:0) T β ( − i h ∇ ) − V (ˆ x ) − Ch r − (cid:1) θ r,k (cid:3) − ≥ M X k =1 Tr (cid:2) θ r (cid:0) T β ( − i h ∇ ) − z k | ˆ x | − − Cδ h − (cid:1) θ r (cid:3) − , where we have written θ r ( x ) = θ − ( | x | /r ). We have used here that Cr − + C ≤ Cr − + C ≤ Cδ h − . (85)It is this relation which sets a lower bound on δ . We will control the error using the combinedDaubechies-Lieb-Yau inequality in Theorem 16 with m = h − and α = β / h − ( ≤ HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 25 that mα − = β − / h − ≥ h − . Thus using Theorem 16 we find, for all density matrices γ and all ε ≥ δ , that ε Tr (cid:2) γ (cid:0) θ r ( T β ( − i h ∇ ) − z k | ˆ x | − − Cε − δ h − ) θ r (cid:1)(cid:3) ≥ − C ( εδ − / + ε − / δ + ε − δ ) h − . Thus for all density matrices γ and all ε ≥ δ we have Cδ h − Tr[ γθ r ] ≤ ε Tr (cid:2) γθ r ( T β ( − i h ∇ ) − z k | ˆ x | − ) θ r (cid:3) + C ( εδ − / + ε − / δ + ε − δ ) h − . (86)Hence M X k =1 Tr (cid:2) θ r,k (cid:0) T β ( − i h ∇ ) − V (ˆ x ) − Ch r − (cid:1) θ r,k (cid:3) − (87) ≥ (1 − ε ) M X k =1 Tr (cid:2) θ r ( T β ( − i h ∇ ) − z k | ˆ x | − ) θ r (cid:3) − − C ( εδ − / + ε − / δ + ε − δ ) h − . For the corresponding semi-classical expression we have from (6) and (85) (using δ < / | x − − y − | ≤ | x − y | ) that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 πh ) − Z φ − ( v ) (cid:2) p − V ( v ) (cid:3) − dvdp − M X k =1 (2 πh ) − Z θ r ( v ) (cid:2) p − z k | v | − (cid:3) − dvdp (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cδ / h − . (88)A simple rescaling applied to the local Hydrogen result in Lemma 25 gives that (cid:12)(cid:12)(cid:12)(cid:12) Tr (cid:2) θ r ( T β ( − i h ∇ ) − z k | ˆ x | − ) θ r (cid:3) − − (2 πh ) − Z θ r ( v ) (cid:2) p − z k | v | − (cid:3) − dvdp − z k h − S ( β / h − z k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ch − ( h − r ) − / = Ch − δ / . (89)Combining (87), (88), and (89), using that S is a bounded function (since it is non-increasing;see Lemma 25), that δ < / 2, and that(2 πh ) − Z θ r ( v ) (cid:2) p − z k | v | − (cid:3) − dvdp ≤ Ch − r / = Ch − δ − / , and choosing ε = δ , we conclude thatTr (cid:2) φ − (cid:0) T β ( − i h ∇ ) − V (ˆ x ) − Ch r − (cid:1) φ − (cid:3) − ≥ (2 πh ) − Z φ − ( v ) (cid:2) q − V ( v ) (cid:3) − dvdq + h − M X k =1 z k S ( β / h − z k ) − Cδ / h − . (90) Control of the second term in (83) . Here we use the local rescaled semi-classics in Theorem 23. Before we apply our semi-classical estimates on the support of Φ − φ + we localise using the functions ϕ u from (60) forgeneral M and with ℓ ( u ) as in (56), with ℓ = r/ 4. From (77) and the choice of δ it followsthat ℓ < h small enough. If x is in the support of Φ − φ + and in the support of ϕ u then d ( u ) > r/ ℓ since (using (58)) r ≤ d ( x ) ≤ d ( u ) + ℓ ( u ) < d ( u ) + max { d ( u ) , ℓ } , and also d ( u ) ≤ R + 1 since ℓ ( u ) < / 2. Using again the relativistic IMS localisation (23)we thus haveΦ − φ + (cid:0) T β ( − i h ∇ ) − V (ˆ x ) − Ch r − χ { d ( x ) ≤ r } − Ch R − (cid:1) φ + Φ − = Z r/ 2) and, by (61) (valid on the supportof ϕ u when d ( u ) > r/ ℓ ), r − χ { d ( x ) ≤ r } ( x ) ϕ u ( x ) ≤ Cℓ ( u ) − ϕ u ( x ) . This way, we have proved thatTr (cid:2) Φ − φ + (cid:0) T β ( − i h ∇ ) − V (ˆ x ) − Ch r − χ { d ( x ) ≤ r } − Ch R − (cid:1) φ + Φ − (cid:3) − (94) ≥ Z r/ Note in particular that βf ( u ) ≤ βd ( u ) − ≤ β/r = 2 βδh − ≤ δ ≤ u with r/ ≤ d ( u ) ≤ R + 1, (cid:12)(cid:12)(cid:12) Tr (cid:2) φ + ϕ u (cid:0) T β ( − i h ∇ ) − V (ˆ x ) − Ch ℓ ( u ) − (cid:1) ϕ u φ + (cid:3) − − (2 πh ) − Z φ + ( v ) ϕ u ( v ) (cid:2)p β − q + β − − β − − V ( v ) − Ch ℓ ( u ) − (cid:3) − dvdq (cid:12)(cid:12)(cid:12) ≤ Ch − / f ( u ) − / ℓ ( u ) − / . (96)The semi-classical integral may be estimated using (72) (cid:12)(cid:12)(cid:12)(cid:12)Z (cid:2)p β − q + β − − β − − V ( v ) − Ch ℓ ( u ) − (cid:3) − dq − Z (cid:2) q − V ( v ) (cid:3) − dq (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ch (cid:0) | V ( v ) | + h ℓ ( v ) − (cid:1) / + Ch ℓ ( v ) − (cid:0) | V ( v ) | + h ℓ ( v ) − (cid:1) / ≤ C (cid:0) h | V ( v ) | / + h ℓ ( v ) − | V ( v ) | / + ( hℓ ( v ) − ) (cid:1) , (97)for v in the support of ϕ u , since then we have ℓ ( v ) ≤ ℓ ( u ) / | V ( v ) | ≤ Cd ( v ) − ≤ Cd ( u ) − ≤ Ch − (see (5), (61) and (58)). We have also used that β ≤ h andthat, by (58) and (77) ( ℓ = r/ h ℓ ( v ) − ≤ Ch r − = Cδ h − ≤ Ch − .Combining (94), (96), and (97) (remembering that d ( u ) ≤ Cd ( v ) if v is in the support of ϕ u and d ( u ) > r/ ℓ ) we find, using (60), (5), and (95), thatTr (cid:2) Φ − φ + (cid:0) T β ( − i h ∇ ) − V (ˆ x ) − Ch r − χ { d ( x ) ≤ r } − Ch R − (cid:1) φ + Φ − (cid:3) − ≥ (2 πh ) − Z φ + ( v ) (cid:2) q − V ( v ) (cid:3) − dq dv − C Z r/ Step 2: Upper bound on the quantum energy. We obtain an upper bound on the quantum energy by choosing the density matrix γ = M X k =1 θ r,k γ k θ r,k + Z d ( u ) < R +1 φ + ϕ u γ u ϕ u φ + ℓ ( u ) − du , (101)where γ k , k = 1 , . . . , M , are the density matrices γ k = χ (cid:0) θ r,k (cid:0) T β ( − i h ∇ ) − z k | ˆ x − r k | − (cid:1) θ r,k (cid:1) and γ u , u ∈ R , are the density matrices given in Theorem 23 for the potential V with B ℓ being the ball centered at u , ℓ = ℓ ( u ), f = f ( u ) (see (95)), and φ = φ + ϕ u . Since M X k =1 θ r,k + φ = φ − + φ = 1 , (102)we immediately see from (60) that γ is a density matrix.Using this density matrix as a trial state we obtain from Theorem 23 thatTr (cid:2) T β ( − i h ∇ ) − V (ˆ x ) (cid:3) − ≤ M X k =1 Tr (cid:2) θ r,k (cid:0) T β ( − i h ∇ ) − V (ˆ x ) (cid:1) θ r,k (cid:3) − + (2 πh ) − Z d ( u ) < R +1 φ + ( v ) ϕ u ( v ) (cid:2)p β − q + β − − β − − V ( v ) (cid:3) − ℓ ( u ) − dvdqdu + Ch − / Z r/ Together with (88), (89), (102), and (103) this gives the proof of (10), and therefore finishesthe proof of (7). Step 3: Properties of the density. We will now show that the density matrix γ in (101) satisfies the two requirements (8)and (9).The density of γ is ρ γ ( x ) = M X k =1 θ r,k ( x ) ρ k ( x ) + Z d ( u ) < R +1 ϕ u ( x ) φ ( x ) ρ u ( x ) ℓ ( u ) − du , (104)where ρ k for k = 1 , . . . , M is the density of the density matrix γ k and ρ u for u ∈ R is thedensity for γ u . We first control the 6 / θ r,k ρ k . If β / h − < / α = β / h − ≤ / ν = 2 z k , and m = h − to obtain that0 ≥ Tr (cid:2) θ r,k γ k θ r,k ( T β ( − i h ∇ ) − z k | ˆ x − r k | − ) (cid:3) ≥ Tr (cid:2) θ r,k γ k θ r,k T β ( − i h ∇ ) (cid:3) − Cz / k h − − Ch − z / k r / − Cz k h ≥ Tr (cid:2) θ r,k γ k θ r,k T β ( − i h ∇ ) (cid:3) − Ch − δ − / , where the constant C depends on z k . Hence we have thatTr (cid:2) T β ( − i h ∇ ) θ r,k γ k θ r,k (cid:3) ≤ Ch − δ − / = Ch − / . (105)Using (14) with α = β / h − ≤ m = h − , (105) implies that Z ( θ r,k ρ k ) / ≤ Ch − / Z h ( θ r,k ρ k ) / ≤ β − / h h ( θ r,k ρ k ) / ! / r / + C Z h ( θ r,k ρ k ) / >β − / h β − / h ( θ r,k ρ k ) / ! / r / ≤ Ch − / h − / h / + Ch − / h / ≤ Ch − / , (106)where we have used that r = h and that h is bounded above by a constant. Likewise we find Z θ r,k ρ k ≤ Ch − / . The case when 1 / ≤ β / h − ≤ r − = h from the nucleus z k differently. Let e θ ± ( x ) = θ ± ( | x − r k | /h ).Using the relativistic IMS formula (Theorem 13) and Theorem 14 with ℓ = h / m = h − , α = β / h − , and Ω = {| x − r k | < h } we find that0 ≥ Tr (cid:2) θ r,k γ k θ r,k ( T β ( − i h ∇ ) − z k | ˆ x − r k | − ) (cid:3) ≥ Tr (cid:2)e θ − γ k e θ − ( T β ( − i h ∇ ) − z k | ˆ x − r k | − − Ch − ) (cid:3) + Tr (cid:2) θ r,k e θ + γ k θ r,k e θ + ( T β ( − i h ∇ ) − z k | ˆ x − r k | − − h − χ Ω ) (cid:3) − Ch − . To treat the first term we use the inequality (see (21)) √− ∆ − π | ˆ x | ≥ A s ( − ∆) s − B s , which holds for all 0 ≤ s < / A s , B s > s . Hence,using that h is bounded above by a constant and that 1 ≤ β − / h ≤ ≥ Tr (cid:2)e θ − γ k e θ − ( T β ( − i h ∇ ) − z k | ˆ x − r k | − − Ch − ) (cid:3) ≥ Tr (cid:2)e θ − γ k e θ − ( A s ( − ∆) s − C s h − ) (cid:3) . We appeal to the standard (Daubechies)-Lieb-Thirring inequalityTr (cid:2) ( − ∆) s e θ − γ k e θ − (cid:3) ≥ c Z ( e θ − ρ k ) (3+2 s ) / , which holds for all s ∈ (0 , < s < / (cid:2)e θ − γ k e θ − ( T β ( − i h ∇ ) − z k | ˆ x − r k | − − Ch − ) (cid:3) ≥ c Z ( e θ − ρ k ) (3+2 s ) / − Ch − Z ( e θ − ρ k ) ≥ ( c/ Z ( e θ − ρ k ) (3+2 s ) / − Ch (4 s − /s . Using the Daubechies inequality (Theorem 9) we find as above thatTr (cid:2) θ r,k e θ + γ k θ r,k e θ + ( T β ( − i h ∇ ) − z k | ˆ x − r k | − − h − χ Ω ) (cid:3) ≥ c Tr (cid:2) θ r,k e θ + γ k θ r,k e θ + T β ( − i h ∇ ) (cid:3) − Ch − / . By choosing s sufficiently close to 1 / h is bounded by a constant we concludethat 0 ≥ c Z ( e θ − ρ k ) (3+2 s ) / + c Tr (cid:2) θ r,k e θ + γ k θ r,k e θ + T β ( − i h ∇ ) (cid:3) − Ch − / . As above it follows from this, choosing s sufficiently close to 1 / 2, that we still have Z ( θ r,k ρ k ) / ≤ Ch − / , Z θ r,k ρ k ≤ Ch − / . (107)Using that r = h and that from (5) | V ( x ) | ≤ Cd ( x ) − we also have Z ( h − θ r,k | V − | / ) / ≤ Ch − / , Z h − θ r,k | V − | / ≤ Ch − / . (108)We move to the second term in (104). By the rescaled semi-classics (Theorem 23) we haveon the support of ϕ u φ + that (for f ( u ), see (95)) (cid:12)(cid:12)(cid:12) ρ u ( x ) − / (3 π ) − h − | V ( x ) − | / (cid:12)(cid:12)(cid:12) ≤ Ch − − / f ( u ) / ℓ ( u ) − / + Ch − | V ( x ) − | / , where we have used that on the support of ϕ u φ + we have | V ( x ) | ≤ Cd ( u ) − ≤ Cr − ≤ Ch − ≤ Chβ − , since d ( u ) ≥ r/ ϕ u φ + is non-vanishing. We moreover have on thesupport of ϕ u φ + that | V ( x ) − | / ≤ Cf ( u ) ≤ Cf ( u ) / ℓ ( u ) − / . For r/ < d ( u ) ≤ ℓ ( u ) − ≥ d ( u ) − = f ( u ) ≥ f ( u ) (see (61)) and for d ( u ) > ℓ ( u ) ≤ f ( u ) ≤ 1. Hence (cid:13)(cid:13) ϕ u φ (cid:0) ρ γ − / (3 π ) − h − | V − | / (cid:1)(cid:13)(cid:13) / ≤ Ch − − / f ( u ) / ℓ ( u ) / , (109) HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 31 Using (101) and (102), we have that (cid:13)(cid:13) ρ γ − / (3 π ) − h − | V − | / (cid:13)(cid:13) / (110) ≤ M X k =1 (cid:0) k θ r,k ρ k k / + Ch − k θ r,k | V − | / k / (cid:1) + Z d ( u ) < R +1 (cid:13)(cid:13) ϕ u φ (cid:0) ρ u − / (3 π ) − h − | V − | / (cid:1)(cid:13)(cid:13) / ℓ ( u ) − du + Z d ( u ) > R +1 Ch − (cid:13)(cid:13) ϕ u φ | V − | / (cid:13)(cid:13) / ℓ ( u ) − du . The last term is non-zero only in the case µ = 0 in which case it is easily seen by (6) and(77) to be bounded by Ch − / . Thus, combining (106)–(109), (110) implies that (cid:13)(cid:13) ρ γ − / (3 π ) − h − | V − | / (cid:13)(cid:13) / ≤ Ch − + Ch − − / Z C − r Let S : [0 , /π ] → R be thefunction from Lemma 25. Then there exists a constant C > such that, for all α ∈ [0 , /π ] and κ ∈ (0 , , (cid:12)(cid:12)(cid:12) Tr (cid:2)p − α − ∆ + α − − α − − | ˆ x | − + κ (cid:3) − − (2 π ) − Z (cid:2) p − | v | − + κ (cid:3) − dpdv − S ( α ) (cid:12)(cid:12)(cid:12) ≤ Cκ / . (111) Here, as before, √− α − ∆ + α − − α − = − ∆ / , when α = 0 .Proof of Lemma 27. A simple rescaling, changing x → κ − πx/ 2, givesTr (cid:2)p − α − ∆ + α − − α − − | ˆ x | − + κ (cid:3) − = κ Tr (cid:2)p − β − h ∆ + β − − β − − π | ˆ x | + 1 (cid:3) − , where β = κα and h = 2 κ / /π . We have β ≤ h .The semi-classical integral may be rewritten in the same fashion,(2 π ) − Z (cid:2) p − | v | − + κ (cid:3) − dpdv = κ (2 πh ) − Z (cid:2) p − π | v | + 1 (cid:3) − dpdv . Since the potential V ( x ) = π | x | − C > (cid:12)(cid:12)(cid:12) Tr (cid:2)p − β − h ∆ + β − − β − − π | ˆ x | + 1 (cid:3) − − (2 πh ) − Z (cid:2) p − π | v | + 1 (cid:3) − dpdv − h − π S ( α ) (cid:12)(cid:12)(cid:12) ≤ C h − / . Using that h = 2 κ / /π gives (111). (cid:3) We can now, using the alternative characterization of the function S in Lemma 27, finishthe proof of Theorem 4. Step 4: Continuity of the function S . We recall that T β ( p ) = ( p β − p + β − − β − , β > p , β = 0 . (112)It suffices to prove continuity ofTr (cid:2) T α ( − i ∇ ) − | ˆ x | − + κ (cid:3) − = Tr (cid:2)p − α − ∆ + α − − α − − | ˆ x | − + κ (cid:3) − at all α ∈ [0 , /π ], for any κ ∈ (0 , 1] fixed. Then continuity of S follows from (111) byuniform convergence as κ → α = 0.Let χ > = χ | p |≥ λ , χ < = χ | p |≤ λ for some λ > − Γ )(Γ − Γ ) ∗ ≥ Γ ∗ + Γ Γ ∗ ≤ Γ Γ ∗ + Γ Γ ∗ . Using this with Γ = ε / χ < | ˆ x | − / ,Γ = ε − / χ > | ˆ x | − / for some ε > T α (ˆ p ) − | ˆ x | − + κ (113) ≥ χ > (cid:0) T α (ˆ p ) − (1 + ε − ) | ˆ x | − + κ (cid:1) χ > + χ < (cid:0) T α (ˆ p ) − (1 + ε ) | ˆ x | − + κ (cid:1) χ < . Here and in the sequel we write T α (ˆ p ) for the operator T α ( − i ∇ ) (and similarly for otheroperators). Since T α ≥ T α for α ≤ α , and T α ( p ) ≥ α − | p | − α − , (113) implies that, if HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 33 α ∈ (0 , A ] for some A > 0, then for all ε > T α (ˆ p ) − | ˆ x | − + κ ≥ χ > (cid:0) A − | ˆ p | − A − − (1 + ε − ) | ˆ x | − + κ (cid:1) χ > + χ < (cid:0) T α (ˆ p ) − (1 + ε ) | ˆ x | − + κ (cid:1) χ < . (114)Since | ˆ p | − / ( π | ˆ x | ) ≥ 0, we have that12 A − | ˆ p | − (1 + ε − ) | ˆ x | − ≥ , if A ≤ ε/ (2 π ), and now assuming ε ≤ 1. Furthermore, for λ ≥ A − we have that χ > (cid:0) A − | ˆ p | − A − (cid:1) χ > ≥ . This implies that, if ε ≤ λ ≥ A − , α ∈ (0 , A ] and A ≤ ε/ (2 π ), then by (114) T α (ˆ p ) − | ˆ x | − + κ ≥ χ < (cid:0) T α (ˆ p ) − (1 + ε ) | ˆ x | − + κ (cid:1) χ < . (115)Since, by a Taylor-expansion, T α ( p ) ≥ T ( p ) − ( αp ) / 8, and since χ < = χ | p |≤ λ , we havethat, still for α ∈ (0 , A ], T α (ˆ p ) − | ˆ x | − + κ ≥ χ < (cid:0) T (ˆ p ) − α λ / − (1 + ε ) | ˆ x | − + κ (cid:1) χ < . (116)Let γ α,κ = χ ( −∞ , (cid:0) T α (ˆ p ) − | ˆ x | − + κ (cid:1) . Then (116) and the fact that T ≥ T α imply that, for α ∈ (0 , A ],Tr (cid:2) T (ˆ p ) − | ˆ x | − + κ (cid:3) − ≥ Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − = Tr (cid:2) γ α,κ (cid:0) T α (ˆ p ) − | ˆ x | − + κ (cid:1)(cid:3) ≥ Tr (cid:2) γ α,κ χ < (cid:0) T (ˆ p ) − α λ / − (1 + ε ) | ˆ x | − + κ (cid:1) χ < (cid:3) . (117)If κ ∈ (0 , α ∈ (0 , A ], λ ≥ A − , and A ≤ / (2 π ) we will show the a priori estimateTr (cid:2) γ α,κ χ < (cid:3) ≤ Cκ − / and Tr (cid:2) γ α,κ χ < | ˆ x | − χ < (cid:3) ≤ Cκ − / . (118)The combined Daubechies-Lieb-Yau inequality (32) gives that for positive constants C , C such that α ≤ / ( C π ), we haveTr (cid:2) T α (ˆ p ) − C | ˆ x | − + C κ (cid:3) − ≥ − Cα / − C Z | x | Choose λ = 2 A − , A = ǫ/ (2 π ). We combine (117) and (118) and use the variationalprinciple to conclude that for α ∈ (0 , ǫ/ (2 π )], ǫ < 1, and κ ∈ (0 , (cid:2) T (ˆ p ) − | ˆ x | − + κ (cid:3) − ≥ Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − ≥ Tr (cid:2)(cid:0) χ < γ α,κ χ < (cid:1)(cid:0) T (ˆ p ) − | ˆ x | − + κ (cid:1)(cid:3) − Cκ − / ( α ε − + ε ) ≥ Tr (cid:2) T (ˆ p ) − | ˆ x | − + κ (cid:3) − − Cκ − / ( α ε − + ε ) . Finally choose ǫ = α / ; then α ≤ (2 π ) − / implies that α ∈ (0 , ǫ/ (2 π )] and ǫ < α ≤ (2 π ) − / and κ ∈ (0 , (cid:2) T (ˆ p ) − | ˆ x | − + κ (cid:3) − ≥ Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − ≥ Tr (cid:2) T (ˆ p ) − | ˆ x | − + κ (cid:3) − − Cκ − / α / , which proves continuity from the right of Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − at α = 0 for any κ ∈ (0 , κ .)We now prove the continuity at any α ∈ (0 , /π ). Note first that, for 0 < α ≤ α , T α ( p ) ≥ T α ( p ) ≥ ( α − α ) T α ( p ) . (119)Assume first that α > α , and let γ α,κ be defined as above. Then, using (119) and thevariational principle,Tr[ T α (ˆ p ) − | ˆ x | − + κ (cid:3) − ≤ Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − ≤ Tr (cid:2) γ α,κ (cid:0) T α (ˆ p ) − | ˆ x | − + κ (cid:1)(cid:3) ≤ Tr (cid:2) γ α,κ (cid:0) T α (ˆ p ) − | ˆ x | − + κ (cid:1)(cid:3) + [( αα − ) − 1] Tr (cid:2) γ α,κ T α (ˆ p ) (cid:3) = Tr[ T α (ˆ p ) − | ˆ x | − + κ (cid:3) − + [( αα − ) − 1] Tr (cid:2) γ α,κ T α (ˆ p ) (cid:3) . It remains to show that [( αα − ) − 1] Tr (cid:2) γ α,κ T α (ˆ p ) (cid:3) → α → α . For this, it obviouslysuffices to show that Tr (cid:2) γ α,κ T α (ˆ p ) (cid:3) is uniformly bounded for, say, α ∈ ( α , A ] for some A ∈ ( α , /π ). But this follows as in the proof of (118). This proves continuity from theright of Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − at α ∈ (0 , /π ). To prove continuity from the left, assume α < α , and let γ α ,κ be defined as above. Then, by (119) and the variational principle,Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − ≥ Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − = Tr (cid:2) γ α ,κ (cid:0) T α (ˆ p ) − | ˆ x | − + κ (cid:1)(cid:3) = Tr (cid:2) γ α ,κ (cid:0) T α (ˆ p ) − | ˆ x | − + κ (cid:1)(cid:3) + Tr (cid:2) γ α ,κ ( T α (ˆ p ) − T α (ˆ p )) (cid:3) ≥ Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − + [1 − ( α α − ) ] Tr (cid:2) γ α ,κ T α (ˆ p ) (cid:3) . As before, the last trace is finite by arguments as in the proof of (118) (since α < /π ).This proves continuity from the left, and therefore, continuity, of Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − at α ∈ (0 , /π ).Finally we prove the continuity at α = 2 /π . Here, arguments as in the proof of (118)are no longer at our disposal. Therefore, let ǫ > 0, and let γ α ,κ be defined as above, andchoose φ , . . . , φ N ∈ C ∞ ( R ), ( φ i , φ j ) = δ i,j , such thatTr (cid:2) γ N (cid:0) T α (ˆ p ) − | ˆ x | − + κ (cid:1)(cid:3) (120) ≤ Tr (cid:2) γ α ,κ (cid:0) T α (ˆ p ) − | ˆ x | − + κ (cid:1)(cid:3) + ǫ/ (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − + ǫ/ , for γ N ( x, y ) = P Nj =1 φ j ( x ) φ j ( y ). This is possible since the operator is defined as theFriedrichs extension from C ∞ ( R ). (Here, both N and the φ j ’s depend, of course, on ǫ ). HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 35 Recall that γ N is finite dimensional and φ j ∈ C ∞ ( R ). Using this, (119), and the variationalprinciple gives that (for any α ∈ ( α / , α )),Tr (cid:2) γ N (cid:0) T α (ˆ p ) − | ˆ x | − + κ (cid:1)(cid:3) = Tr (cid:2) γ N T α (ˆ p ) (cid:3) + Tr (cid:2) γ N (cid:0) − | ˆ x | − + κ (cid:1)(cid:3) ≥ Tr (cid:2) γ N (cid:0) T α (ˆ p ) − | ˆ x | − + κ (cid:1)(cid:3) + [( α − α ) − 1] Tr (cid:2) γ N T α (ˆ p ) (cid:3) ≥ Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − + [( α − α ) − 1] Tr (cid:2) γ N T α / (ˆ p ) (cid:3) . (121)Choose now δ > α ∈ ( α − δ, α ) ∩ ( α / , α ) ⇒ [( α − α ) − 1] Tr (cid:2) γ N T α / (ˆ p ) (cid:3) > − ǫ/ . (122)Then, combining (120), (121), and (122) (and using (119) again) we have proved that, forall ǫ > < δ < α / α ∈ ( α − δ, α ) ⇒ Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − ≥ Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − ≥ Tr (cid:2) T α (ˆ p ) − | ˆ x | − + κ (cid:3) − − ǫ . This proves the continuity from the left of Tr (cid:2) T α (ˆ p ) −| ˆ x | − + κ (cid:3) − at α = 2 /π , and thereforefinishes the proof that S : [0 , /π ] → R is continuous.This completes the proof of Theorem 4. (cid:3) Local relativistic semi-classical estimates using new coherent states In this section we study the sum and the density of the negative eigenvalues of the localisedHamiltonian φH β φ , with φ compactly supported and H β = T β ( − i h ∇ ) + V (ˆ x ). Here, T β isgiven by (3), and V is a (sufficiently) regular potential (see below for details). For the mostpart we suppress the index β but all estimates, in particular the constants C, will be uniformin β ∈ [0 , G u,q .Let 1 /a > h > 0. The kernel of G u,q is given by G u,q ( x, y ) = ( πh ) − n/ e − a ( x + y − u ) +i q ( x − y ) /h − h a ( x − y ) . (123)A first important property of these operators is their completeness. Lemma 28 ( Completeness of new coherent states). The coherent operators G u,q satisfy Z G u,q dq (2 πh ) n = G b (ˆ x − u ) , Z G u,q du (2 πh ) n = G b ( − i h ∇ − q ) , (124) where ˆ x denotes the operator multiplication by the position variable x . Here G b ( v ) =( b/π ) n/ e − bv with b = 2 a/ (1 + h a ) . Note that G b has integral 1 and hence Z G u,q dudq (2 πh ) n = . (125)We shall consider operators of the form Z G u,q f ( b A u,q ) G u,q dudq , (126)where f : R → R is any polynomially bounded real function. As we shall see in the nexttheorem the integrand above is a trace class operator for each ( u, q ). The integral above isto be understood in the weak sense, i.e., as a quadratic form. We shall consider situationswhere the integral defines bounded or unbounded operators. Theorem 29 ( Trace identity). Let f : R → R and V : R n → R be polynomially bounded,real-valued measurable functions and let ˆ A = B + B ˆ x − i hB ∇ be a first order self-adjoint differential operator with B ∈ R , B , ∈ R n . Then G u,q f ( ˆ A ) G u,q V (ˆ x ) is a trace class operator (when extended from C ∞ ( R n ) ) and Tr (cid:2) G u,q f ( ˆ A ) G u,q V (ˆ x ) (cid:3) = Z f ( B + B v + B p ) G b ( u − v ) G b ( q − p ) G ( h b ) − ( z ) × V ( v + h ab ( u − v ) + z ) dvdpdz . In particular, Tr (cid:2) G u,q (cid:3) = 1 . We shall also need the following extension of this theorem, where we however only givean estimate on the trace. Theorem 30 ( Trace estimates). Let f, ˆ A be as in the previous theorem. Let moreover φ ∈ C n +4 ( R n ) be a bounded, real function with all derivatives up to order n + 4 bounded,and let V, F ∈ C ( R n ) be real functions with bounded second derivatives. Then, for h < , < a < /h and b = 2 a/ (1 + h a ) we have, with σ ( u, q ) = F ( q ) + V ( u ) , that Tr (cid:2) G u,q f ( ˆ A ) G u,q φ (ˆ x ) (cid:0) F ( − i h ∇ ) + V (ˆ x ) (cid:1) φ (ˆ x ) (cid:3) = Z f ( B + B v + B p ) G b ( u − v ) G b ( q − p ) × h(cid:0) φ ( v + h ab ( u − v )) + E ( u, v ) (cid:1) σ ( v + h ab ( u − v ) , p + h ab ( q − p ))+ E ( u, v ; q, p ) i dvdp , with k E k ∞ , k E k ∞ ≤ Ch b , where C depends only on sup | ν |≤ n +4 k ∂ ν φ k ∞ , sup | ν | =2 k ∂ ν V k ∞ , and sup | ν | =2 k ∂ ν F k ∞ . (Note that the assumption < a < /h implies < b < /h .) We will use the above theorem to prove an upper bound on the sum of eigenvalues of theoperator F ( − i h ∇ ) + V (ˆ x ), in the case when F ( q ) = T β ( q ) from (3) with β ∈ [0 , 1] (equalto p β − q + β − − β − for β ∈ (0 , q when β = 0). This is done in Lemma 34below by constructing a trial density matrix on the form (126).To prove a lower bound on the sum of the negative eigenvalues one approximates theHamiltonian F ( − i h ∇ ) + V (ˆ x ) by an operator also represented on the form (126). Theorem 31 ( Coherent states representation). Consider functions F, V ∈ C ( R n ) ,for which all second and third derivatives are bounded. Let σ ( u, q ) = F ( q ) + V ( u ) , then for a < /h and b = 2 a/ (1 + h a ) we have the representation (as quadratic forms on C ∞ ( R n ) ), F ( − i h ∇ ) + V (ˆ x ) = Z G u,q b H u,q G u,q dudq (2 πh ) n + E , with the operator-valued symbol b H u,q = σ ( u, q ) + b ∆ σ ( u, q ) + ∂ u σ ( u, q )(ˆ x − u ) + ∂ q σ ( u, q )( − ih ∇ − q ) . (127) The operator ˆ A is essentially self-adjoint on Schwartz functions on R n . The operator G u,q f ( ˆ A ) G u,q φ (ˆ x ) ( F ( − i h ∇ ) + V (ˆ x )) φ (ˆ x ) is originally defined on, say, C ∞ ( R n ), but it ispart of the claim of the theorem that it extends to a trace class operator on all of L ( R n ). HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 37 The error term, E , is a bounded operator with k E k ≤ Cb − / X | ν | =3 k ∂ ν σ k ∞ + Ch b X | ν | =2 k ∂ ν σ k ∞ . Let us recall our convention that x − = min { x, } and that χ = χ ( −∞ , denotes thecharacteristic function of ( −∞ , Theorem 32 ( Local relativistic semi-classics). For n ≥ , let φ ∈ C n +40 ( R n ) besupported in a ball B ⊂ R n of radius and let V ∈ C ( B ) be a real function. Let ≤ β ≤ , h > , and let σ β ( u, q ) = T β ( q ) + V ( u ) and H β = T β ( − i h ∇ ) + V (ˆ x ) with T β ( q ) = p β − q + β − − β − for β ∈ (0 , and T ( q ) = q .Then (cid:12)(cid:12)(cid:12) Tr (cid:2) φH β φ (cid:3) − − (2 πh ) − n Z φ ( u ) σ β ( u, q ) − dudq (cid:12)(cid:12)(cid:12) ≤ Ch − n +6 / . The constant C > here depends only on k φ k C n +4 , k V k C , and the dimension n , but noton β ∈ [0 , . The important property for our method to work is that the second and third orderderivatives of the kinetic energy function T β ( q ) are bounded uniformly in q and β . Thusthe error term above is independent of β ∈ [0 , − h ∆ / V , which corresponds to the limit β → 0. We prove upperand lower bounds and start with the lower bound. Lemma 33 ( Lower bound on Tr[ φH β φ ] − ). Under the same conditions as in Theorem32, Tr[ φH β φ ] − ≥ (2 πh ) − n Z φ ( u ) σ β ( u, q ) − dudq − Ch − n +6 / . The constant C > here depends only on k φ k C n +4 , k V k C , and the dimension n , but noton β ∈ [0 , .Proof. Since φ has support in the ball B we may assume without loss of generality that V ∈ C ( R n ) with the support in a ball B of radius 2 and that the norm k V k C refers tothe supremum over all of R n . We shall not explicitly follow how the error terms dependon k φ k C n +4 and k V k C . All constants denoted by C depend on k φ k C n +4 , k V k C , and thedimension n but, in particular, not on β .We use the Daubechies inequality (Theorem 9) to control various error estimates. Since T β ( q ) ≥ T ( q ) for β ∈ [0 , 1] we may use it with β = 1. Then, uniformly in β ∈ [0 , φH β φ ] − ≥ C k φ k ∞ Z u ∈ B σ ( u, q ) − dudq (2 πh ) n ≥ − Ch − n . Consider some fixed 0 < τ < h and β ). If h ≥ τ then we get thatTr[ φH β φ ] − ≥ Z φ ( u ) σ β ( u, q ) − dudq (2 πh ) n − Cτ − / h − n +6 / . We are therefore left with considering h < τ . We use the convention that k ψ k C p = sup | ν |≤ p k ∂ ν ψ k ∞ . If we now use the inequality [ x + y ] − ≥ [ x ] − + [ y ] − , which we will do frequently withoutfurther mentioning, and Theorem 31, we have thatTr[ φH β φ ] − ≥ Tr (cid:20)Z φ G u,q b H ( ε ) u,q G u,q φ dudq (2 πh ) n (cid:21) − + Tr (cid:2) φ (cid:0) ε p − β − h ∆ + β − − εβ − − C ( b − / + h b ) (cid:1) φ (cid:3) − . (128)Here, 0 < ε < / b H ( ε ) u,q = e σ ( u, q ) + b ∆ e σ ( u, q ) + ∂ u e σ ( u, q )(ˆ x − u ) + ∂ q e σ ( u, q )( − ih ∇ − q )with e σ ( u, q ) = (1 − ε ) T β ( q ) + V ( u ). The second trace can be estimated from below usingthe Daubechies inequality (Theorem 9) with α = β / h − , m = h − . ThenTr (cid:2) φ (cid:0) ε p − β − h ∆ + β − − εβ − − C ( b − / + h b ) (cid:1) φ (cid:3) − = ε Tr (cid:2) φ (cid:0)p − β − h ∆ + β − − β − − Cε − ( b − / + h b ) (cid:1) φ (cid:3) − ≥ − Cεh − n Z B (cid:0) ε − ( b − / + h b ) (cid:1) n/ dx − Cεβ n/ h − n Z B (cid:0) ε − ( b − / + h b ) (cid:1) n dx . (129)We shall eventually choose ε = ( b − / + h b ). Note that then ε < / 2, and that the boundin (129) is − Ch − n ( b − / + h b ), uniformly for β ∈ [0 , φH β φ ] − ≥ Z Tr h φ G u,q (cid:2) b H ( ε ) u,q (cid:3) − G u,q φ i dudq (2 πh ) n − Ch − n ( b − / + h b ) . We first consider the integral over u outside the ball B of radius 2, where V = 0. UsingTheorem 29 (with f ( t ) = [ t ] − , and V replaced by φ ) and R φ ≤ C , we get that this partof the integral is(1 − ε ) Z u B h T β ( q ) + ( n + ( n − βq ) / [4 b (1 + βq ) / ] + q · ( p − q ) / p βq i − × G b ( q − p ) G b ( u − v ) G ( h b ) − ( z ) φ ( v + h ab ( u − v ) + z ) dvdpdz dudq (2 πh ) n ≥ (1 − ε ) Z z ∈ B φ ( z ) Z u B G b ( u − v ) G ( h b ) − ( v + h ab ( u − v ) − z ) dudvdz × Z (cid:2) T β ( q ) + q · ( p − q ) / p βq (cid:3) − G b ( q − p ) dqdp (2 πh ) n = (1 − ε ) Z z ∈ B φ ( z ) Z (1 − h ab ) u B − v G b ( u ) G ( h b ) − ( v − z ) dudvdz × Z (cid:2) T β ( q ) + q · p/ p βq (cid:3) − G b ( p ) dqdp (2 πh ) n . The integration over u, v is obviously bounded by 1. In fact, the u -integration can be shownto be exponentially small, i.e., less than C e − Cb , but this will not be necessary. HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 39 The domain of integration for the variables q, p is contained in the set { ( q, p ) | | q | ≤ | p |} .Then, Z (cid:2) T β ( q ) + q · p/ p βq (cid:3) − G b ( p ) dqdp ≥ − C Z | q | < | p | | q || p | p βq G b ( p ) dqdp ≥ − C Z | p | n +2 G b ( p ) dp = − C b − ( n +2) / . It follows that the integral over u B is bounded from below by − Ch − n b − / , since b > u ∈ B we use Theorem 29 as before. This time, expanding φ tosecond order in z at the point z = 0 and using the crucial fact (which we shall use withoutmentioning later) that, for any λ > Z x j G λ ( x ) dx = 0 , Z | x | m G λ ( x ) dx = C λ − m/ , (130)implies thatTr[ φH β φ ] − ≥ Z u ∈ B (cid:2) φ ( v + h ab ( u − v )) + Ch b (cid:3) G b ( u − v ) G b ( q − p ) × h H ( ε ) u,q ( v, p ) i − dudq (2 πh ) n dvdp − Ch − n ( b − / + h b ) , (131)where H ( ε ) u,q ( v, p ) = e σ ( u, q ) + b ∆ e σ ( u, q ) + ∂ u e σ ( u, q )( v − u ) + ∂ q e σ ( u, q )( p − q ) . The rest of the proof is simply an estimate of the integral in (131). This analysis is anelementary but tedious exercise in calculus. For the convenience of the reader it is given indetail in Appendix B below. (cid:3) Lemma 34 ( Construction of a trial density matrix). Under the same conditions asin Theorem 32 there exists a density matrix γ on L ( R n ) such that Tr (cid:2) φ ( T β ( − i h ∇ ) + V (ˆ x )) φγ (cid:3) ≤ Z φ ( u ) σ β ( u, q ) − dudq (2 πh ) n + Ch − n +6 / . (132) Moreover, the density ρ γ of γ satisfies (cid:12)(cid:12)(cid:12) ρ γ ( x ) − (2 πh ) − n ω n | V − | n/ (2 + β | V − | ) n/ ( x ) (cid:12)(cid:12)(cid:12) ≤ Ch − n +9 / , (133) for (almost) all x ∈ B and (cid:12)(cid:12)(cid:12)(cid:12)Z φ ( x ) ρ γ ( x ) dx − (2 πh ) − n ω n Z φ ( x ) | V − | n/ (2 + β | V − | ) n/ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ch − n +6 / , (134) where ω n is the volume of the unit ball B in R n . The constants C > in the above estimatesdepend only on n, k φ k C n +4 , and k V k C , but not on β ∈ [0 , . It is convenient to introduce the function η ( t ) = n Z ∞ χ [ T β ( p ) + t ] | p | n − d | p | = | t − | n/ (2 + β | t − | ) n/ . (135)(Recall that χ is the characteristic function of R − .) Proof. We will occasionally drop the index β in H β and σ β . It is important to realize,however, that all estimates are uniform in β . We first note that since T ( p ) ≤ T β ( p ) ≤ T ( p ) = p / | p | − C ≤ σ ( v, p ) ≤ p + C. (136)Let us start by choosing some fixed 0 < τ < 1. For h ≥ τ and for some C > Z φ ( u ) σ ( u, q ) − dudq (2 πh ) n + Cτ − / h − n +6 / ≥ , and that for any s > 0, (2 πh ) − n η ( V ( x )) ≤ Cτ − s h − n + s . If h ≥ τ we may therefore use γ = 0, and s = 9 / 10 and s = 6 / h < τ and, if necessary, that τ is small enoughdepending only on φ and V . Also, as for the lower bound, we may assume that V ∈ C ( R n )with support in the ball B / concentric with B and of radius 3 / u, q ) anoperator ˆ h u,q byˆ h u,q = (cid:26) σ ( u, q ) + b ∆ σ ( u, q ) + ∇ σ ( u, q ) · (ˆ x − u, − i h ∇ − q ) if u ∈ B u B . The corresponding function is h u,q ( v, p ) = (cid:26) σ ( u, q ) + b ∆ σ ( u, q ) + ∇ σ ( u, q ) · ( v − u, p − q ) if u ∈ B u B . As for the lower bound we shall choose a = h − / ; then a < h − . In fact, we will assumethat (1 − h ab ) ≥ / 2. Recall here that b = 2 a/ (1 + h a ) (i.e., in particular a ≤ b ≤ a ).Similar to (172) (for ε = 0) we have for u ∈ B that (cid:12)(cid:12) h u,q ( v, p ) − σ ( v, p ) − ξ v,p ( u − v, q − p ) (cid:12)(cid:12) ≤ C | u − v | ( b − + | u − v | ) + C | q − p | ( b − + | q − p | ) , (137)where ξ v,p ( u, q ) = b ∆ σ ( v, p ) − X i,j ∂ i ∂ j T β ( p ) q i q j − X i,j ∂ i ∂ j V ( v ) u i u j . Recalling that χ is the characteristic function of R − we define γ = Z G u,q χ (cid:2) ˆ h u,q (cid:3) G u,q dudq (2 πh ) n . (138)Since ≤ χ (cid:2) ˆ h u,q (cid:3) ≤ it follows from (125) that ≤ γ ≤ .We now calculate Tr[ γφH β φ ] = Tr (cid:2) γφ ( T β ( − i h ∇ ) + V (ˆ x )) φ (cid:3) . From Theorem 30 we havethatTr (cid:2) γφ ( T β ( − i h ∇ ) + V (ˆ x )) φ (cid:3) = Z χ [ h u,q ( v, p )] G b ( u − v ) G b ( q − p ) h E ( u, v ; q, p ) + (139) (cid:0) φ ( v + h ab ( u − v )) + E ( u, v ) (cid:1) σ ( v + h ab ( u − v ) , p + h ab ( q − p )) i dudq (2 πh ) n dvdp , HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 41 where E , E are functions such that k E k ∞ + k E k ∞ ≤ Ch b . The rest of the proof of(132) is a tedious, but elementary analysis of this integral. A detailed analysis is presentedin Appendix B below.It remains to estimate the density ρ γ together with R φ ( x ) ρ γ ( x ) dx . By Theorem 29 and(138), γ is easily seen to be a trace class operator with density ρ γ ( x ) = Z χ (cid:2) h u + v,q + p ( v, p ) (cid:3) G b ( u ) G b ( q ) G ( h b ) − ( x − v − h abu ) dvdp dudq (2 πh ) n . (140)The proof of (133) and (134) again relies on a detailed analysis of this integral. As forthe estimate on the energy above this analysis is an exercise in calculus. Although it isstill elementary this analysis is more complicated than in the case of the energy. For theconvenience of the reader the analysis is given in detail in Appendix B below. (cid:3) Appendix A. Various Proofs In this appendix we collect proofs of various results mentioned in Section 2. Proof of Theorem 11 ( Operator inequality critical Hydrogen ) . Let f ∈ S ( R ) and t > π Z R | f ( x ) | | x | dx = 1 π Z R Z R ˆ f ( p ) ˆ f ( q ) | p − q | (cid:16) | p | + | p | t | q | + | q | t (cid:17) / (cid:16) | q | + | q | t | p | + | p | t (cid:17) / dpdq ≤ π Z R Z R | ˆ f ( p ) | | p − q | | p | + | p | t | q | + | q | t dpdq . (141)We first compute the integral in q . Since ( | q | + | q | t ) − ≤ | q | − − | q | t − + | q | t − we get Z R | p − q | | q | + | q | t dq ≤ Z R | p − q | ( | q | − − | q | t − + | q | ) t − dq . Note [19, 5.10 (3)] that, for 0 < τ, σ < n , with 0 < τ + σ < n , Z R n | y − z | τ − n | z | σ − n dz = c n − τ − σ c τ c σ c τ + σ c n − τ c n − σ | y | τ + σ − n , (142)where c τ = π − τ/ Γ( τ / n = 3, then Z R | y − z | − | z | − r dz = k r | y | − r for r ∈ (1 , , (143)with k r = π Γ (cid:0) r − (cid:1) Γ (cid:0) − r (cid:1) Γ (cid:0) − r (cid:1) Γ (cid:0) r (cid:1) . (144)It follows that, for 3 < t < Z R | p − q | | p | + | p | t | q | + | q | t dq ≤ k | p | + ( k − k − t ) | p | t − (145)+ ( k − t − k − t (cid:1) | p | t − + k − t | p | t − . We see from (144) that k is symmetric with respect to r = 2. Using Γ(1 + z ) = z Γ( z ) inthe denominator in (144) with z = 1 − r/ z )Γ(1 − z ) = π/ sin( πz ) (for0 < z < 1) in the denominator and numerator we obtain k r = − π tan( πr/ − r/ , which shows that k is decreasing on (1 , 2) and increasing on (2 , t > / 3, we find, for positive constants A ( t − / , B ( t − / , that Z R | p − q | | p | + | p | t | q | + | q | t dq ≤ k | p | − π A ( t − / | p | t − + π B ( t − / . (146)Since k = π , this and (141) implies that2 π Z R | f ( x ) | | x | dx ≤ Z R | ˆ f ( p ) | (cid:0) | p | − A ( t − / | p | t − + B ( t − / (cid:1) dp , (147)which implies the operator inequality, for all t ∈ (5 / , √− ∆ − π | ˆ x | ≥ A ( t − / ( − ∆) ( t − / − B ( t − / . (148)Choosing t = 2 s + 1 proves (21) for s ∈ (1 / , / s ∈ [0 , / A ( τ − / , B ( τ − / , given τ ∈ [1 , / , t ∈ (5 / , 2) and positiveconstants A ( t − / , B ( t − / , such that A ( t − / | p | t − − B ( t − / ≥ A ( τ − / | p | τ − − A ( τ − / . (cid:3) Integral representation for the relativistic kinetic energy. We shall here give aself-contained presentation of the integral formulas for the relativistic kinetic energy. Therelativistc kinetic energy will be given in terms of the modified Bessel functions of the secondkind, K ν . To identify the modified Bessel functions we use that [1, 9.6.23] K ( t ) = Z ∞ e − wt √ w − dw , t > , (149)and the recursion relation [1, 9.6.28] K ν +1 ( t ) = − t ν ddt ( t − ν K ν ( t )) , t > . (150)We emphasise that we use these properties only as definitions of the Bessel functions, andderive all other properties of these functions that we need. Note that K ν : R + → R aresmooth functions.Consider the function G mn ∈ L ( R n ) (the Yukawa potential) whose Fourier transform is b G mn ( ξ ) = (2 π ) − n/ ( | ξ | + m ) − . Using that v − = R ∞ e − uv du we get from the Fourier transform of Gaussian functions thefollowing integral representation for G , G mn ( z ) = Z ∞ (4 πu ) − n/ e − m u −| z | / (4 u ) du . (151)It follows from this that G is non-negative, smooth for z = 0, and indeed in L ( R n ).For odd n the above integral can be explicitly calculated. For even n it is as we shall nowsee expressible as a modified Bessel function K ν of integer order ν . By a simple change ofvariables (2 w = v + v − with v = 2 mu/ | z | ) in the integral (149) we see from (151) that G m ( z ) = (2 π ) − K ( m | z | ) . From the recursion formula (150) we then find inductively thatfor even n G mn ( z ) = m ( n − / (2 π ) − n/ | z | − ( n − / K ( n − / ( m | z | ) . (152) HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 43 In fact, the same formula holds for all n , but we do not wish to discuss the modified Besselfunctions of fractional order (one could simply take this formula as their definition). Lemma 35. The heat kernel for the operator √− ∆ + m on L ( R n ) is given by exp( − t p − ∆ + m )( x, y )= − ∂ t G mn +1 ( x − y, t )= 2 (cid:16) m π (cid:17) ( n +1) / t ( | x − y | + t ) ( n +1) / K ( n +1) / ( m ( | x − y | + t ) / ) for t > .Proof. It suffices to show that the two tempered distributions on R n +1 , t | t | exp( −| t | p − ∆ + m )( x, 0) and − ∂ t G mn +1 ( x, t ) , have the same Fourier transform. The Fourier transform as a function of ξ = ( p, s ) with p ∈ R n and s ∈ R of the first distribution is(2 π ) − ( n +1) / (cid:0) − Z −∞ e − i ts + t √ p + m dt + Z ∞ e − i ts − t √ p + m dt (cid:1) = − s (2 π ) − ( n +1) / | p | + s + m . The Fourier transform of the second distribution above is − s b G mn +1 ( p, s ) = − s (2 π ) − ( n +1) / | p | + s + m . The last identity in the lemma follows from (150) and (152). (cid:3) If we set x = y in the above lemma we find the following integral formula for the modifiedBessel function K ( n +1) / ( t ) = 12 (cid:18) t π (cid:19) ( n − / Z R n e − t √ p +1 dp , t > . For n = 3 this simplifies to K ( t ) = t Z ∞ e − t √ s +1 s ds (153)from which we immediately get the estimate K ( t ) ≤ Ct − e − t/ . (154) Proof of Theorem 13 ( Relativistic IMS formula ) . By scaling, it suffices to prove thestatement for α = 1. We start from the identity (cid:0) f, ( p − ∆ + m − m ) f (cid:1) = Z | f ( x ) − f ( y ) | F ( x − y ) dxdy with F ( x − y ) = m π K ( m | x − y | ) | x − y | , (155) where K is the modified Bessel function of second order defined above (see (149)–(150)).The identity follows from Lemma 35 (for a proof, see [19, 7.12]). Then, (cid:0) f, ( p − ∆ + m − m ) f (cid:1) = Z | f ( x ) − f ( y ) | F ( x − y ) dxdy = Z Z M (cid:2) θ u ( x ) | f ( x ) | + θ u ( y ) | f ( y ) | (cid:3) F ( x − y ) dµ ( u ) dxdy + Z Z M (cid:2) − ( θ u ( x ) + θ u ( y ) ) + θ u ( x ) θ u ( y ) − θ u ( x ) θ u ( y ) (cid:3) × (cid:2) f ( x ) f ( y ) + f ( x ) f ( y ) (cid:3) F ( x − y ) dµ ( u ) dxdy = Z Z M | θ u ( x ) f ( x ) − θ u ( y ) f ( y ) | F ( x − y ) dµ ( x ) dxdy + Z Z M (cid:2) − ( θ u ( x ) + θ u ( y ) ) + θ u ( x ) θ u ( y ) (cid:3) × (cid:2) f ( x ) f ( y ) + f ( x ) f ( y ) (cid:3) F ( x − y ) dµ ( u ) dxdy = Z M (cid:0) θ u f, ( p − ∆ + m − m ) θ u f (cid:1) dµ ( u ) − Z Z M (cid:0) θ u ( x ) − θ u ( y ) (cid:1) f ( x ) f ( y ) F ( x − y ) dµ ( u ) dxdy . This proves (23) with L given by (24)–(25). We now show that k L θ u k ≤ Cm − k∇ θ u k ∞ forfixed u . By (25), Young’s inequality, and (154), (cid:12)(cid:12) ( f, L θ u f ) (cid:12)(cid:12) ≤ m π k∇ θ u k ∞ Z | f ( x ) | | f ( y ) | K ( m | x − y | ) dxdy ≤ C m k∇ θ u k ∞ k f k Z ∞ t K ( mt ) dt = C m − k∇ θ u k ∞ k f k . (156)This proves that L θ u is a bounded operator. (cid:3) Proof of Theorem 14 ( Localisation error ) . Again, by scaling, it suffices to prove the state-ment for α = 1. With χ Ω the characteristic function of Ω (and L ≡ L θ ) we have from therepresentation (25) of L , since θ is constant on Ω c , that L = χ Ω Lχ Ω + (1 − χ Ω ) Lχ Ω + χ Ω L (1 − χ Ω ) . (157)If Γ , Γ are bounded operators, then (Γ − Γ )(Γ − Γ ) ∗ ≥ Γ ∗ + Γ Γ ∗ ≤ Γ Γ ∗ + Γ Γ ∗ . Using this with Γ = ε / χ Ω , Γ = ε − / (1 − χ Ω ) L for some ε > L ≤ χ Ω Lχ Ω + εχ Ω + ε − (1 − χ Ω ) L (1 − χ Ω ) . (158)To bound the first term on the right side recall that k L k ≤ Cm − k∇ θ k ∞ (see (156)). HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 45 Let us now look at the third term in (158). Since θ is constant on Ω c anddist(Ω c , supp ∇ θ ) ≥ ℓ , using (25) givesTr (cid:2) (1 − χ Ω ) L (1 − χ Ω ) (cid:3) = Z x ∈ Ω c ,y ∈ Ω , | x − y | >ℓ L ( x, y ) dxdy ≤ Cm k∇ θ k ∞ Z x ∈ Ω c ,y ∈ Ω , | x − y | >ℓ K ( m | x − y | ) dxdy . Using (154), Z x ∈ Ω c ,y ∈ Ω , | x − y | >ℓ K ( m | x − y | ) dxdy ≤ C e − mℓ Z x ∈ Ω c ,y ∈ Ω , | x − y | >ℓ ( m | x − y | ) − dxdy = C e − mℓ | Ω | Z ∞ ℓ ( mt ) − t dt = Cm − ( mℓ ) − e − mℓ | Ω | . This gives the boundTr (cid:2) (1 − χ Ω ) L (1 − χ Ω ) (cid:3) ≤ Cℓ − k∇ θ k ∞ e − mℓ | Ω | . Finally, we choose ε = m − k∇ θ k ∞ . Then by the above the two first terms in (158) arebounded by Cm − k∇ θ k ∞ χ Ω , and the trace of the third term (which we denote Q θ ) isbounded by Cmℓ − e − mℓ k∇ θ k ∞ | Ω | . (cid:3) For the proof of the combined Daubechies-Lieb-Yau inequality (Theorem 16) we need thefollowing inequality [5]. Lemma 36. For f ∈ S ( R ) , Z R e − m π − | x | | x | | f ( x ) | dx ≤ π √ − (cid:0) f, ( p − ∆ + m − m ) f (cid:1) . (159) Proof. Let µ = m π − . Then I = Z R e − µ | x | | x | | f ( x ) | dx = 12 π Z R Z R ˆ f ( p ) 1 | p − p | ˆ g ( p ) dp dp , with g ( x ) = f ( x )e − µ | x | . Writing ˆ g ( p ) explicitly as the convolution with the Fourier trans-form of e − µ | x | and then applying the Schwarz inequality we get that I ≤ π ( πµ ) / Z R Z R Z R | ˆ f ( p ) | e −| q | / (4 µ ) | p − p − q | | p | | p | dp dp dq . Since [19, 5.10 (3)] Z R | p − p − q | | p | dp = π | p − q | , we have I ≤ π / µ / Z R Z R | ˆ f ( p ) | e −| q | / (4 µ ) | p − q | | p | dp dq . By Newton’s theorem [19, 9.7 (5)], Z R e −| q | / (4 µ ) | p − q | dq = 1 | p | Z | q | < | p | e −| q | / (4 µ ) dq + Z | q | > | p | e −| q | / (4 µ ) | q | dq = 8 πµ | p | Z | p | e − r / (4 µ ) dr ≤ πµ min (cid:8) , ( πµ ) / | p | (cid:9) . Substituting µ = m π − we find that I ≤ π m Z R | ˆ f ( p ) | min {| p | , m | p |} dp , from which the claim follows since √ t + 1 − ≥ ( √ − 1) min { t , t } for t ≥ (cid:3) Proof of Theorem 16 ( Combined Daubechies-Lieb-Yau ) . We may assume that W ( x ) ≤ W by W − .Assume first that να ≤ / (16 πM ). By the Daubechies inequality (17),Tr (cid:2)p − α − ∆ + m α − − mα − + W (ˆ x ) (cid:3) − ≥ Tr (cid:2)p − α − ∆ + m α − − mα − + 2 W χ { d R ( x ) <αm − } (cid:3) − (160) − Cm / Z d R ( x ) >αm − | W ( x ) | / dx − Cα Z d R ( x ) >αm − | W ( x ) | dx . The assumption on the positions of the nuclei implies that χ { d R ( x ) <αm − } = P Mj =1 χ {| x − R j | <αm − } , and so, using the assumption on W , we obtainTr (cid:2)p − α − ∆ + m α − − mα − + 2 W χ { d R ( x ) <αm − } (cid:3) − ≥ M M X j =1 Tr (cid:2)p − α − ∆ + m α − − mα − − (cid:0) νM | ˆ x − R j | + CνM mα − (cid:1) χ {| x − R j | <αm − } (cid:3) − = Tr (cid:2)p − α − ∆ + m α − − mα − − (cid:0) νM | ˆ x | − + CνM mα − (cid:1) χ {| x | <αm − } (cid:3) − . (161)The last equality follows from the translation invariance of − ∆. By scaling,Tr (cid:2)p − α − ∆ + m α − − mα − − (cid:0) νM | ˆ x | − + CνM mα − (cid:1) χ {| x | <αm − } (cid:3) − = α − Tr (cid:2)p − ∆ + m − m − (cid:0) γ | ˆ x | − + Cγm (cid:1) χ {| x | HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 47 and so Tr (cid:2)p − ∆ + m − m − (cid:0) γ | ˆ x | − + Cγm (cid:1) χ {| x | 2, (1 − θ ) / ∈ C ( R ), and k∇ θ k ∞ ≤ Cα − m, k∇ (1 − θ ) / k ∞ ≤ Cα − m . Let θ j ( x ) = θ ( x − R j ), j = 1 , . . . , M , and θ M +1 ( x ) = (1 − P Mj =1 θ j ) / (the latter is well-defined due to the assumption min k = ℓ | R k − R ℓ | > αm − ). The relativistic IMS formulaand the localisation estimate (26) used for Ω j , j = 1 , . . . , M , being the balls centered at R j with radii 3 αm − / ℓ = αm − / 4, and Ω M +1 being the (disjoint) union of the sameballs and ℓ = αm − / 8, gives the operator inequality p − α − ∆ + m α − − mα − + W (ˆ x ) ≥ θ M +1 (cid:0)p − α − ∆ + m α − − mα − − Cmα − M X j =1 χ Ω j + W (ˆ x ) (cid:1) θ M +1 (165)+ M X j =1 θ j (cid:0)p − α − ∆ + m α − − mα − − Cmα − + W (ˆ x ) (cid:1) θ j − M +1 X j =1 Q j , with Tr (cid:2) Q j (cid:3) ≤ C mα − . Here we have used that θ i χ Ω j θ i = δ ij θ i , i, j ∈ { , . . . , M } , θ M +1 χ Ω M +1 θ M +1 = 0, and θ i χ Ω M +1 θ i ≤ θ i , i = M + 1.Using the Daubechies inequality on the first term in (165) and the assumption on W in thesecond (noticing that θ j ( x ) /d R ( x ) = θ j ( x ) / | x − R j | due to the assumption min k = ℓ | R k − R ℓ | > αm − ), we get from this thatTr[ p − α − ∆ + m α − − mα − + W (ˆ x )] − ≥ − Cm / Z d R ( x ) >αm − / | W ( x ) | / dx − Cα Z d R ( x ) >αm − / | W ( x ) | dx − C mα − − C M X j =1 (cid:0) m / ( mα − ) / + α ( mα − ) (cid:1)(cid:12)(cid:12) { x | αm − < | x − R j | < αm − } (cid:12)(cid:12) (166)+ M X j =1 Tr (cid:2) θ j (cid:0)p − α − ∆ + m α − − mα − − Cmα − − ν | ˆ x − R j | − Cνmα − (cid:1) θ j (cid:3) − . By the translation invariance of − ∆, the last term equals M Tr (cid:2) θ (cid:0)p − α − ∆ + m α − − Cmα − − Cνmα − − ν | ˆ x | − (cid:1) θ (cid:3) − , and using the Lieb-Yau inequality (and the properties of θ and that να ≤ /π ),Tr (cid:2) θ (cid:0)p − α − ∆ + m α − − Cmα − − Cνmα − − ν | ˆ x | − (cid:1) θ (cid:3) − ≥ α − Tr (cid:2) θ (cid:0) √− ∆ − π | ˆ x | − − Cmα − (cid:1) θ (cid:3) − ≥ − Cmα − . (167)Further, by the assumption on W , and the assumption min k = ℓ | R k − R ℓ | > αm − , m / Z αm − / Also note that for the Coulomb energy, D (Φ s , Φ s ) = s − D (Φ , Φ) = Cs − . Therefore, weimmediately get that X ≤ i 1. Then g r ( x ) = d r ( x ) − . Using that |∇ V TF | ( x ) ≤ Cg r ( x ) d r ( x ) − = d r ( x ) − we obtain (cid:12)(cid:12) ∇ ρ TF ∗ | x | − (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∇ (cid:16) M X j =1 z j | x − r j | − − V TF ( x ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ M X j =1 z j | x − r j | − + d r ( x ) − ≤ M max j { z j } (cid:0) min j | x − r j | (cid:1) − + d r ( x ) − = Cd r ( x ) − + d r ( x ) − ≤ Cg r ( x ) . If on the other hand d r ( x ) < 1, then, by using (36) and (41), (cid:12)(cid:12) ∇ ρ TF ∗ | x | − (cid:12)(cid:12) ≤ Z (cid:12)(cid:12) ∇ ρ TF ( y ) (cid:12)(cid:12) | x − y | − dy ≤ C Z g r ( y ) d r ( y ) − | x − y | − dy = C M X j =1 Z d r ( y )= | y − r j | g r ( y ) d r ( y ) − | x − y | − dy . Since g r ( y ) ≤ d r ( y ) − / = | y − r j | − / when d r ( y ) = | y − r j | this implies that (cid:12)(cid:12) ∇ ρ TF ∗ | x | − (cid:12)(cid:12) ≤ C M X j =1 Z | y − r j | − / | x − y | − dy = C M X j =1 | x − r j | − / ≤ CM d r ( x ) − / = Cg r ( x ) . This finishes the proof of (169).Let us now proceed to prove inequality (43). That the difference is positive is again justsuperharmonicity of | x | − . It is easy to see that | d r ( x ) − d r ( y ) | ≤ | x − y | . (170)In the case when d r ( x ) ≥ t we can conclude that ρ TF ∗ | x | − − ρ TF ∗ | x | − ∗ Φ t ≤ sup | z − x |≤ t (cid:8) (cid:12)(cid:12) ∇ ρ TF ∗ | z | − (cid:12)(cid:12) (cid:9) Z | x − y | Φ t ( x − y ) dy ≤ Ct sup | z − x |≤ t g r ( z ) ≤ Ctg r ( x ) . In the last step we have used that if d r ( x ) ≥ t and | z − x | ≤ t (this condition stems fromthe support of Φ t ), then inequality (170) guarantees that d r ( x ) ≤ d r ( z ) ≤ d r ( x ). This inturn implies ( ) − g r ( x ) ≤ g r ( z ) ≤ ( ) − / g r ( x ).If, on the other hand, d r ( x ) ≤ t , then we claim that (cid:12)(cid:12) ρ TF ∗ | x | − − ρ TF ∗ | y | − (cid:12)(cid:12) ≤ C | x − y | / . (171)This can be seen as follows. (cid:12)(cid:12) ρ TF ∗ | x | − − ρ TF ∗ | y | − (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z ddθ (cid:16) ρ TF ∗ | θx + (1 − θ ) y | − (cid:17) dθ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z ∇ (cid:16) ρ TF ∗ | θx + (1 − θ ) y | − (cid:17) · ( x − y ) dθ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z g r ( θx + (1 − θ ) y ) | x − y | dθ ≤ C | x − y | M X j =1 Z | θx + (1 − θ ) y − r j | − / dθ = C | x − y | / M X j =1 Z (cid:12)(cid:12)(cid:12)(cid:12) θ ( x − y ) | x − y | + y − r j | x − y | (cid:12)(cid:12)(cid:12)(cid:12) − / dθ . Let n = ( x − y ) / | x − y | , b = ( y − r j ) / | x − y | , and c = n · b . Then | θn + b | ≥ | θ + n · b | = ( θ + c ) .Therefore (cid:12)(cid:12) ρ TF ∗ | x | − − ρ TF ∗ | y | − (cid:12)(cid:12) ≤ C | x − y | / M X j =1 Z | θ + c | − / dθ . The integral R | θ + c | − / dθ is bounded uniformly for c ∈ R . This proves (171). HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 51 This allows us finally to show that for d r ( x ) ≤ t , ρ TF ∗ | x | − − ρ TF ∗ | x | − ∗ Φ t = Z (cid:0) ρ TF ∗ | x | − − ρ TF ∗ | y | − (cid:1) Φ t ( x − y ) dy ≤ C Z | x − y | / Φ t ( x − y ) dy = Ct / . This finishes the proof of the corollary. (cid:3) Appendix B. Estimates of semi-classical integrals In this appendix we give the remaining arguments on the analysis of the integrals in thesemi-classical proofs of Lemma 33 and Lemma 34. Proof of Lemma 33 ( Lower bound on Tr[ φH β φ ] − ): Estimate of integral (131) . It re-mains to estimate the integral in (131). Note that by Taylor’s formula for e σ we have H ( ε ) u,q ( v, p ) ≥ e σ ( v, p ) + e ξ v,p ( u − v, q − p ) − C | u − v | ( b − + | u − v | ) (172) − C | q − p | ( b − + | q − p | ) , where e ξ v,p ( u, q ) = b ∆ e σ ( v, p ) − (1 − ε ) X i,j ∂ i ∂ j T β ( p ) q i q j − X i,j ∂ i ∂ j V ( v ) u i u j . We have used that | ∆ e σ ( v, p ) − ∆ e σ ( u, q ) | ≤ C | u − v | + C | q − p | , and similarly, when replacing ∂ i ∂ j F ( q ) by ∂ i ∂ j F ( p ), and ∂ i ∂ j V ( u ) by ∂ i ∂ j V ( v ). We get that: H ( ε ) u,q ( v, p ) ≤ ⇒ | p | ≤ C (cid:0) | u − v | + | q − p | (cid:1) . (173)(Note that this holds also for ε = 0, and uniformly in β ∈ [0 , T β ( p ) ≥ T ( p ).) This implies that Z H ( ε ) u,q ( v,p ) ≤ (cid:0) | u − v | m + | q − p | m (cid:1) G b ( u − v ) G b ( q − p ) dpdudq ≤ Cb − m/ , (174)and Z H ( ε ) u,q ( v,p ) ≤ (cid:0) | u − v | m + | q − p | m (cid:1) G b ( u − v ) G b ( q − p ) dpdvdq ≤ Cb − m/ . (175)From this we obtain that Z G b ( u − v ) G b ( q − p ) h H ( ε ) u,q ( v, p ) i − dpdqdv ≥ − C , (176)and hence from (131) thatTr[ φH β φ ] − ≥ Z u ∈ B φ ( v + h ab ( u − v )) G b ( u − v ) G b ( q − p ) [ H u,q ( v, p )] − dudq (2 πh ) n dvdp − Ch − n ( b − / + h b ) . Here we have used the fact that the u -integration is over a bounded region. From now onwe may however ignore the restriction on the u -integration. We note that, by using (173)and (174), that φ has support in B , and that b > 1, we get that Z H ( ε ) u,q ( v,p ) ≤ φ ( v + h ab ( u − v )) (cid:0) | u − v | ( b − + | u − v | ) + | q − p | ( b − + | q − p | (cid:1) × G b ( u − v ) G b ( q − p ) dudqdvdp ≤ Cb − / . Using this and (172) we find after the simple change of variables u → u + v and q → q + p that Tr[ φH β φ ] − ≥ Z φ ( v + h abu ) G b ( u ) G b ( q ) × (cid:2)e σ ( v, p ) + e ξ v,p ( u, q ) − C | u | ( b − + | u | ) − C | q | ( b − + | q | ) (cid:3) − dudq (2 πh ) n dvdp − Ch − n ( b − / + h b ) ≥ Z φ ( v + h abu ) G b ( u ) G b ( q ) (cid:2)e σ ( v, p ) + e ξ v,p ( u, q ) (cid:3) − dudq (2 πh ) n dvdp − Ch − n ( b − / + h b ) . (177)At this point we divide the ( v, p )-integration into three regions given in terms of a parameterΛ > − = { ( v, p ) | e σ ( v, p ) ≤ − Λ } , Ω + = { ( v, p ) | e σ ( v, p ) ≥ Λ } , Ω = { ( v, p ) | | e σ ( v, p ) | < Λ } . The parameter Λ will be chosen such that 1 ≥ Λ ≥ Cb − for some sufficiently largeconstant C . This is possible if τ is small enough and hence b large enough. Then, since allthe second derivatives of e σ are bounded we may assume that b | ∆ e σ ( v, p ) | < Λ / v, p ), uniformly in β .We first consider Ω + . We see from (177) that we only need to integrate over the set { ( u, q ) | C ( | u | + | q | ) ≥ Λ } . Also notice that e σ ( v, p ) ≥ | p | − C (since T β ( p ) ≥ T ( p )) showsthat we only need to integrate over the set { p | | p | ≤ C (1 + | q | + | u | ) } . Therefore, Z ( v,p ) ∈ Ω + φ ( v + h abu ) G b ( u ) G b ( q ) (cid:2)e σ ( v, p ) + e ξ v,p ( u, q ) (cid:3) − dudqdvdp ≥ − C e − Cb Λ . A similar argument shows that on Ω − we can ignore the negative part [ ] − paying thesame price − Ch − n e − Cb Λ .For ( v, p ) ∈ Ω − we estimate the integral Z φ ( v + h abu ) G b ( u ) G b ( q ) (cid:2)e σ ( v, p ) + e ξ v,p ( u, q ) (cid:3) dudq ≥ ( φ ( v ) + Ch b ) e σ ( v, p ) − Ch b . Here we have expanded φ to second order at the point v and used the crucial fact that Z e ξ v,p ( u, q ) G b ( u ) G b ( q ) dudq = 0 . (178) HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 53 For ( v, p ) ∈ Ω − we have, of course, e σ ( v, p ) = e σ ( v, p ) − . Since the volume of Ω − is boundedby a constant we get for the integration over Ω − ∪ Ω + , Z ( v,p ) ∈ Ω − ∪ Ω + φ ( v + h abu ) G b ( u ) G b ( q ) (cid:2)e σ ( v, p ) + e ξ v,p ( u, q ) (cid:3) − dudqdvdp ≥ Z ( v,p ) ∈ Ω − ∪ Ω + φ ( v ) e σ ( v, p ) − dvdp − C ( h b + e − Cb Λ ) . (179)Finally, let ( v, p ) ∈ Ω . Observe that, with ϑ ( t ) = (2 t + βt ) n/ , Z ( v,p ) ∈ Ω dp = c n (cid:0) ϑ ( − [Λ − V ( v )] − ) − ϑ ( − [Λ + V ( v )] − ) (cid:1) ≤ C Λ , (180)by the mean value theorem (uniformly in v ). Now, φ ( v + h abu ) G b ( u ) G b ( q ) (cid:2)e σ ( v, p ) + e ξ v,p ( u, q ) (cid:3) − ≥ φ ( v + h abu ) G b ( u ) G b ( q ) (cid:2)e σ ( v, p ) (cid:3) − − Cφ ( v + h abu ) G b ( u ) G b ( q )( b − + | u | + | q | ) , and, using the observation above and making the change of variables v → v − h abu in the v -integral, Z ( v,p ) ∈ Ω φ ( v + h abu ) G b ( u ) G b ( q )( b − + | u | + | q | ) dvdpdudq ≤ C Λ b − . Expanding φ to first order at v we have that Z ( v,p ) ∈ Ω φ ( v + h abu ) G b ( u ) G b ( q ) e σ ( v, p ) − dvdpdudq ≥ Z ( v,p ) ∈ Ω φ ( v ) e σ ( v, p ) − dvdp + Ch ab Z ( v,p ) ∈ Ω ,v ∈ supp V | u | G b ( u ) G b ( q ) e σ ( v, p ) − dudqdvdp ≥ Z ( v,p ) ∈ Ω φ ( v ) e σ ( v, p ) − dvdp − Chb / Λ . As a consequence, Z ( v,p ) ∈ Ω φ ( v + h abu ) G b ( u ) G b ( q ) (cid:2)e σ ( v, p ) + e ξ v,p ( u, q ) (cid:3) − dudqdvdp ≥ Z ( v,p ) ∈ Ω φ ( v ) e σ ( v, p ) − dvdp − C Λ(Λ hb / + b − ) . (181) Since Z φ ( v ) e σ ( v, p ) − dvdp = (1 − ε ) − n Z φ ( v ) (cid:2)p β − p + (1 − ε ) β − − (1 − ε ) β − + V ( v ) (cid:3) − dvdp ≥ (1 − ε ) − n Z φ ( v ) σ β ( v, p ) − dvdp ≥ Z φ ( v ) σ β ( v, p ) − dvdp − Cε , the lemma follows from (177), (179), and (181) if we choose b = h − / . (cid:3) Proof of Lemma 34 ( Construction of a trial density ): Estimates of integrals . Wegive here the remaining arguments on the analysis of the integrals in the semi-classicalproofs of Lemma 34. The energy: proof of (132) . It remains to estimate the integral in (139).Using (173), that T β ( p ) ≤ p , and that h u,q ( v, p ) = 0 unless u ∈ B , we get that Z χ [ h u,q ( v, p )] G b ( u − v ) G b ( q − p ) (cid:0) T β ( p + h ab ( q − p )) (cid:1) dudqdvdp ≤ C . This implies thatTr[ γφH β φ ] ≤ Z χ [ h u,q ( v, p )] G b ( u − v ) G b ( q − p ) φ (cid:0) v + h ab ( u − v ) (cid:1) × σ (cid:0) v + h ab ( u − v ) , p + h ab ( q − p ) (cid:1) dudq (2 πh ) n dvdp + Ch bh − n . From (137) we may now conclude thatTr[ γφH β φ ] ≤ Z u ∈ B − v χ (cid:2) σ ( v, p ) + ξ v,p ( u, q ) − C | u | ( b − + | u | ) − C | q | ( b − + | q | ) (cid:3) G b ( u ) G b ( q ) × φ ( v + h abu ) σ ( v + h abu, p + h abq ) dudq (2 πh ) n dvdp + Ch bh − n . (182)At this point we introduce the same partition of the ( v, p )-integration into sets Ω ± , Ω as inthe proof of the lower bound above (with ε = 0) with the same Λ = b − / = h / .Then for the integration over Ω + we have as above that C ( | u | + | q | ) > Λ and hence Z ( v,p ) ∈ Ω+ ,u ∈ B − v χ (cid:2) σ ( v, p ) + ξ v,p ( u, q ) − C | u | ( b − + | u | ) − C | q | ( b − + | q | ) (cid:3) × φ ( v + h abu ) σ ( v + h abu, p + h abq ) G b ( u ) G b ( q ) dudqdvdp ≤ C e − cb Λ ≤ Ch b , where we have used (136) and that φ is supported in the ball B .Similarily, if ( v, p ) ∈ Ω − then for the ( u, q )-integration we can safely assume that theargument of χ is negative to the effect of paying the same e − Cb Λ price. Likewise we mayignore the restriction u ∈ B − v , since u B − v and v + h abu ∈ B implies | u | > HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 55 (1 − h ab ) − > 1. Expanding φ and σ to second order at ( v, p ) ∈ Ω − and using the factthat all their second order derivatives are bounded together with (130) we get that Z χ (cid:2) σ ( v, p ) + ξ v,p ( u, q ) − C | u | ( b − + | u | ) − C | q | ( b − + | q | ) (cid:3) × φ ( v + h abu ) σ ( v + h abu, p + h abq ) G b ( u ) G b ( q ) dudq ≤ Z h(cid:0) φ ( v ) + h ab u · ∇ ( φ )( v ) (cid:1) (cid:0) σ ( v, p ) + h ab ( u, q ) · ∇ σ ( v, p ) (cid:1)i G b ( u ) G b ( q ) dudq + Ch b + C e − Cb Λ ≤ φ ( v ) σ ( v, p ) − + Ch b . It is important here that σ and ∇ σ are bounded uniformly in β ≤ − . This followsfrom (136) and |∇ σ β ( v, p ) | ≤ C (1 + | p | ). Indeed, (136), in particular, implies that Ω − is abounded set (uniformly in β ). The fact that the volume of Ω − is bounded also gives thatthe contribution from Ω − to the integral on the right side of (182) is bounded above by(2 πh ) − n Z Ω − φ ( v ) σ ( v, p ) − dvdp + Ch bh − n . Finally, we consider ( v, p ) ∈ Ω . If we expand φ to first order at v and σ to second orderat ( v, p ) and use that all second order derivatives of σ are bounded and that ∇ σ ( v, p ) isbounded for ( v, p ) ∈ Ω we obtain that φ ( v + h abu ) σ ( v + h abu, p + h abq ) ≤ φ ( v ) σ ( v, p ) + Ch ab ( | u | + | q | ) + Ch a b ( | u | + | q | ) . This together with the estimate | χ ( x + y ) x − χ ( x ) x | ≤ | y | implies that Z u ∈ B − v χ (cid:2) σ ( v, p ) + ξ v,p ( u, q ) − C | u | ( b − + | u | ) − C | q | ( b − + | q | ) (cid:3) × φ ( v + h abu ) σ ( v + h abu, p + h abq ) G b ( u ) G b ( q ) dudq ≤ Z χ [ σ ( v, p ) + ξ v,p ( u, q ) − C | u | ( b − + | u | ) − C | q | ( b − + | q | )] G b ( u ) G b ( q ) dudq × φ ( v ) σ ( v, p ) + Ch b / ≤ φ ( v ) σ ( v, p ) − + C ( b − + h b / ) . We have here again used that the effect of removing the restriction u ∈ B − v causes asmaller error than the last term above. Note that u ∈ B − v and v + h abu ∈ B imply | u | ≤ − h ab ) − ≤ | v | ≤ | v + u | + | u | < 8. If weuse that (180) implies Vol(Ω ∩ { v | | v | < } × R n ) ≤ C Λwe see that the contribution from Ω to the integral on the right side of (182) is boundedabove by (2 πh ) − n Z Ω φ ( v ) σ ( v, p ) − dvdp + Ch − n ( b − + h b / )Λ . This finishes the proof of the upper bound on the energy in (132) . The density: proof of (133) and (134) . Here it remains to estimate the integral in(140). The strategy is to freeze the variable | p | in ξ v,p so that the remaining dependenceon | p | is explicitly integrable. This is accomplished in the estimate (183) below. After the | p | -integration the proof is almost the same as in the non-relativistic case [36]. We write p = | p | ω and define p = ( β | V ( v ) − | + 2 | V ( v ) − | ) / ω = η ( V ( v )) /n ω . We will then prove that if u ∈ B − v then χ [ σ ( v, p )+ ξ v,p ( u, q )+ R ( u, q )] ≤ χ [ h u + v,q + p ( v, p )] ≤ χ [ σ ( v, p )+ ξ v,p ( u, q ) − R ( u, q )] , (183)where R ( u, q ) = C ( | u | ( b − + | u | ) + ( | q | + Λ)( b − + | q | ) + ( b − + | u | + | q | )( | u | + | q | )Λ − ) . We first prove (183) for ( v, p ) ∈ Ω . In this case we have η ( V ( v ) + Λ) /n ≤ p ≤ η ( V ( v ) − Λ) /n , from which it follows that | p − p | ≤ C Λ with a constant independent of β ∈ [0 , G ( t ) = p β − t + β − − β − , so that T β ( p ) = G ( p ) (we suppress that G depends on β ). Note that then ∂ i ∂ j T β ( p ) = 4 p i p j G ′′ ( p ) + 2 δ ij G ′ ( p ), and so, in particular, ∆ T β ( p ) =4 G ′′ ( p ) p + 2 nG ′ ( p ). Therefore, using that p i = | p | ω i , p ,i = | p | ω i , | ξ v,p ( u, q ) − ξ v,p ( u, q ) |≤ b | ∆ σ ( v, p ) − ∆ σ ( v, p ) | + X i,j (cid:12)(cid:12) ∂ i ∂ j [ T β ( p ) − T β ( p )] (cid:12)(cid:12) | q i q j |≤ C ( b − + | q | ) (cid:0) | G ′′ ( p ) p − G ′′ ( p ) p | + | G ′ ( p ) − G ′ ( p ) | (cid:1) ≤ C ( b − + | q | ) | p − p | ≤ C Λ( b − + | q | ) . (184)Here we have used the choice of p and that G ′ ( t ) and tG ′′ ( t ) have bounded derivativesuniformly in β ∈ [0 , | h u + v,q + p ( v, p ) − σ ( v, p ) − ξ v,p ( u, q ) | ≤ C | u | ( b − + | u | ) + | q | ( b − + | q | ) + C Λ( b − + | q | ) , which is, in fact, stronger than (183).If ( v, p ) ∈ Ω + we see that the left inequality in (183) is only violated if ξ v,p ( u, q ) ≤ − Λand the right inequality is only violated if h u + v,q + p ( v, p ) ≤ 0. Since ( v, p ) ∈ Ω + we must inboth cases have C ( | u | + | q | ) > Λ. We hence get (again using (137)) that (cid:12)(cid:12) h u + v,q + p ( v, p ) − σ ( v, p ) − ξ v,p ( u, q ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) h u + v,q + p ( v, p ) − σ ( v, p ) − ξ v,p ( u, q ) (cid:12)(cid:12) + (cid:12)(cid:12) ξ v,p ( u, q ) (cid:12)(cid:12) + (cid:12)(cid:12) ξ v,p ( u, q ) (cid:12)(cid:12) ≤ C | u | ( b − + | u | ) + C | q | ( b − + | q | ) + C ( b − + | u | + | q | ) ≤ C | u | ( b − + | u | ) + C | q | ( b − + | q | ) + C ( b − + | u | + | q | )( | u | + | q | )Λ − , which gives (183) in this case.Finally, if ( v, p ) ∈ Ω − then the left inequality in (183) is only violated if h u + v,q + p ( v, p ) ≥ ξ v,p ( u, q ) ≥ Λ. In both cases this implies that C ( | u | + | q | ) > Λ and hence the same argument as for Ω + proves (183).Using (183) we can estimate the density in (140) from above and below. We will discussthe lower bound on the density. The upper bound is similar. Performing the | p | -integral in HE RELATIVISTIC SCOTT CORRECTION FOR ATOMS AND MOLECULES 57 (140) we obtain ρ γ ( x ) ≥ Z u ∈ B − v Ξ( u, q, v, ω ) G b ( u ) G b ( q ) G ( h b ) − ( x − v − h abu ) dvdω dudq (2 πh ) n (185)= Z (1 − h ab ) u ∈ B − v Ξ( u, q, v − h abu, ω ) G b ( u ) G b ( q ) G ( h b ) − ( x − v ) dvdω dudq (2 πh ) n , where Ξ( u, q, v, ω ) = n − η ( V ( v ) + ξ v,p ( u, q ) + R ( u, q )) . We have that V ( v − h abu ) + ξ v − h abu,p ( u, q ) + R ( u, q ) ≤ V ( v ) − h abu ∇ V ( v ) + ξ v,p ( u, q ) + R ( u, q ) + Ch a b | u | + Ch ab | u | ( b − + | u | ) ≤ V ( v ) − h abu ∇ V ( v ) + ξ v,p ( u, q ) + R ( u, q ) + Ch a b | u | . (186)(In the last line we have used that h ab ≤ R ( u, q ).) We now use that (cid:12)(cid:12) η ( s ) − η ( t ) − η ′ ( t )( s − t ) (cid:12)(cid:12) ≤ (cid:26) C | s − t | / + C ( | s | + | t | ) | s − t | , n = 3 C ( | s | n − + | t | n − + | s | n − + | t | n − ) | s − t | , n ≥ . (187)We continue with the case n = 3 and leave n ≥ η ′ ( V ( v )) is bounded independently of β ∈ [0 , 1] we obtain from(186) and (187) n Ξ( u, q, v − h abu, ω ) ≥ η ( V ( v )) + η ′ ( V ( v ))( ξ v,p ( u, q ) − h abu ∇ V ( v )) − C (cid:0) b − + | q | + | u | + h ab | u | + R ( u, q ) (cid:1) / − C (cid:0) b − + | q | + | u | + h ab | u | + R ( u, q ) (cid:1) − CR ( u, q ) − Ch a b | u | − Ch ab | u | . (188)It is now crucial that (see (130) and (178)) Z ( ξ v,p ( u, q ) − h abu ∇ V ( v )) G b ( u ) G b ( q ) dudq = 0 . Since v ∈ supp( V ) ⊂ B / and (1 − h ab ) u B − v implies | u | > / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z (1 − h ab ) u ∈ B − v ( ξ v,p ( u, q ) − h abu ∇ V ( v )) G b ( u ) G b ( q ) dudq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C e − b/ ≤ Ch / . (189)Combining this with (188) and inserting it into (185) we arrive at (recall that Λ = b − / ) ρ γ ( x ) ≥ (2 πh ) − ω Z η (cid:2) V ( v ) (cid:3) G ( h b ) − ( x − v ) dv − Ch − (cid:0) h / + b − / + ( h ab ) / b − / (cid:1) . (190)Here we have removed the constraint (1 − h ab ) u ∈ B − v by the same argument as above.We shift the v -coordinate by x , and then expand η (cid:2) V ( x + v ) (cid:3) in the integral at x byexpanding V to second order at x and using (187). Then η (cid:2) V ( x + v ) (cid:3) ≥ η (cid:2) V ( x ) (cid:3) + η ′ (cid:2) V ( x )] ∇ V ( x ) · v − C ( | v | / + | v | ) . Then we obtain from (190) (using (130)) that ρ γ ( x ) − (2 πh ) − ω η (cid:2) V ( x ) (cid:3) ≥ − Ch − (cid:0) h / + ( h b ) / (cid:1) ≥ − Ch − / . This finishes the proof of (133).Lastly, we prove (134). 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