arXiv:2009.02474v1 [math-ph] 5 Sep 2020 otedsrt pcrmof spectrum discrete the to where space rudsaeeeg,nml h n iheulweights equal with one gen whe L¨uders the [11] such to namely according from energy, occurs refrain state that grou we ground state the However, the approximate pick enough. which simply and States be would state. well ground sufficiently a need even not γ For ei ta 8 rvdta h rudsaeeeg sa eigenvalue an is energy state ground the that proved [8] al et Lewis sn neato novn h lcrnssi rsn,w set we present, spin electron’s the involving interaction no is ipiiy oevr eaersrcigorevst eta atom neutral to ourselves restricting are we Moreover, simplicity. h orsodn udai omwt omdomain form with form quadratic corresponding the where nqaiy[,Catr5 qain(.3] tflosta h omi bo is form the if that only follows and it if (5.33)], below Equation 5, Chapter [7, inequality number (1) operator drasekhar = ntefloig eaeitrse npoete fgon ttso states ground of properties in interested are we following, the In Date ipedsrpinehbtn oeqaiaiefaue faoso la of atoms of features qualitative some exhibiting description simple A / < Z/c Z/c ν X N ψ =1 c w etme ,2020. 5, September : 1 ihkntceeg ( energy kinetic with Abstract. td h eairo h n-atcegon tt density state ground one-particle Z the of behavior the study oterltvsi yrgncdniyo hsscale. this on convergenc density the hydrogenic showing relativistic Simon the Barry to and authors the by result µ Z ψ , . . . , eoe h eoiyo ih.I sdfie steFidih extensio Friedrichs the as defined is It light. of velocity the denotes ∈ UETL RN,KNTNI EZ N EN SIEDENTOP HEINZ AND MERZ, KONSTANTIN FRANK, L. RUPERT − EAIITCSRN CT CONJECTURE: SCOTT STRONG RELATIVISTIC p ≥ 2 (0 1 with /π ntelimit the in − , r egt uhthat such weights are 0 2 c t omdmi is domain form its /π M 2 N ∆ .W eoetersligHmloinby Hamiltonian resulting the denote We ). n rudsaeof state ground any ν ecnie ev eta tm faoi number atomic of atoms neutral heavy consider We lcrn n with and electrons + Z/c ,c Z, c 4 ≤ − ∞ → c 2 2 c p C 2 /π 2 − HR PROOF SHORT A Z + keeping ie n rhnra aeo h rudstate ground the of base orthonormal any Given . seas ebt[,Term25 n ee [14]). Weder and 2.5] Theorem [5, Herbst also (see 1. | c x 4 Z H µ X ) M ν =1 Introduction 1 | 1 / / 2 w 2 Z/c − C ( P + µ R q Z c | 1 1 ψ 3 µ M 2 xd egv hr ro farecent a of proof short a give We fixed. pnsae ahi ffrdb h Chan- the by offered is each states spin ≤ N =1 a ewitnas written be can µ sdarayb hnrska.We Chandrasekhar. by already used ν<µ ih X : w ψ C µ ≤ µ q N | N .I h olwn,w would we following, the In 1. = ) | ∩ x V ν V − ν N ν N 1 =1 =1 x w µ C L ntelnt scale length the on | 1 0 2 ∞ ( = C ftedensity the of e R ( in q R Z 3 · · · o notational for 1 = . 3 Z : ν ^ s : N =1 C esrn the measuring n of dsaeenergy state nd modeled = C N q L q .Snethere Since ). hssystem. this f C w .B Kato’s By ). = 2 Z ne from unded eralizations M ( g atomic rge R belonging Z 3 = n fix and : M C of n q − ) 1 . 2 R.L.FRANK,K.MERZ,ANDH.SIEDENTOP
We write dZ for this state. Its one-particle ground state density is M 2 ρZ (x) := N wµ ψµ(x, x2,...,xN ) dx2 dxN . R3(N−1) | | ··· µ=1 X Z For ℓ N0 we denote by Yℓ,m, m = ℓ,...,ℓ, a basis of spherical harmonics of degree∈ℓ, normalized in L2(S2) [12, Formula− (B.93)] and by ℓ (2) Π = Y Y ℓ | ℓ,mih ℓ,m| = mX−ℓ the projection onto the angular momentum channel ℓ. The electron density ρℓ,Z of the L¨uders state in the ℓ-th angular momentum channel is
(3) ρℓ,Z (x) M ℓ N 2 := wµ Yℓ,m(ω)ψµ( x ω, x2,...,xN )dω dx2 dxN . 4π R3(N−1)| S2 | | | ··· µ=1 = X mX−ℓ Z Z We note the relation ∞ (4) ρZ = ρℓ,Z . =0 Xℓ Our main result concerns these densities on distances of order Z−1 from the nucleus. We recall that electrons on these length scales are responsible for the Scott correction in the asymptotic expansion of the ground state energy [13, 3] and are described by the Chandrasekhar hydrogen Hamiltonian √ ∆+1 1 γ x −1 in L2(R3). Spherical symmetry leads to the radial operators − − − | | d2 ℓ(ℓ + 1) γ (5) Cℓ,γ := 2 + 2 +1 1 r−dr r − − r 2 2 H in L (R+) := L (R+, dr). We write ψ , n N0, for a set orthonormal eigen- n,ℓ ∈ functions of Cℓ,γ spanning its pure point spectral space. The hydrogenic density in channel ℓ is ∞ (6) ρH (x) := (2ℓ + 1) ψH ( x ) 2/(4π x 2), x R3, ℓ | n,ℓ | | | | | ∈ n=0 X and the total hydrogenic density is then given by ∞ H H (7) ρ := ρℓ . =0 Xℓ The generalization of Lieb’s strong Scott conjecture [9, Equation (5.37)] to the present situation asserts the convergence of the rescaled ground state densities ρZ H H and ρℓ,Z to the corresponding relativistic hydrogenic densities ρ and ρℓ . Whereas the non-relativistic conjecture was proven by Iantchenko et al [6], it was shown in the present context in [2], including the convergence of the sums defining the limiting H H objects ρℓ and ρ . Here, we will take the existence of the limiting densities for granted. The purpose of this note is to offer a simpler proof of the core of the above convergence of the quantum densities. It is a simple virial type argument which allows us to obtain the central estimate on the difference of perturbed and RELATIVISTIC STRONG SCOTT CONJECTURE 3 unperturbed one-particle Chandrasekhar eigenvalues in an easy way. In addition we will refrain from using the elaborate classes of test functions of [2]. 2 N Theorem 1 (Convergence for fixed angular momentum). Pick γ (0, π ), ℓ0 0, −1/2 −1/2 ∈ 2 ∈ assume Z/c = γ fixed and U : R+ R with C U C B(L (R+)). Then → ℓ,−γ ◦ ◦ ℓ,−γ ∈
−3 −1 H (8) lim c ρℓ0,Z (c x)U( x ) dx = ρℓ0 (x)U( x ) dx. Z→∞ R3 | | R3 | | Z Z We would like to add three remarks: 1. Our hypothesis allows for Coulomb tails of U in contrast to [2, Theorem 1.1] where the test functions were assumed to decay like O(r−1−ε). 2. Since C 0 C0 0 the Sobolev inequality shows that U such that U ℓ, ≥ , ◦|·|∈ L3(R3) L3/2(R3) is allowed. 3. Picking∩ a suitable class of test functions U, we may – by an application of Weierstraß’ criterion – sum (8) and interchange the limit Z with the sum → ∞ over ℓ0. This yields for γ (0, 2/π) and Z/c = γ fixed the convergence of the total density ∈
−3 −1 H (9) lim c ρZ (c x)U( x ) dx = ρ (x)U( x ) dx. Z→∞ R3 | | R3 | | Z Z We refer to [2] for details.
2. Proof of the convergence The general strategy of the proof of Theorem 1 is a linear response argument which was already used by Baumgartner [1] and Lieb and Simon [10] for the con- vergence of the density on the Thomas-Fermi scale and by Iantchenko et al [6] and [2] in the context of the strong Scott conjecture. The rescaled density integrated against a test function U is written by the Hellmann-Feynman theorem as a de- rivative of the energy with respect to perturbing one-particle potential λU and taking the limit Z as the representation − → ∞ Z 1 1 (10) 3 ρℓ,Z (x/c)U( x ) dx = 2 Tr [CZ (CZ λ (Uc Πℓ)ν )]dZ R3 c | | λc { − − ⊗ } ν=1 Z X 2 where Uc(r) := c U(cr). Of course, it is enough to prove (9) for positive U, since we can simply prove the result for the positive and negative part separately and take the difference. By standard estimates following [2] (which in turn are patterned by Iantchenko et al [6]) one obtains
Proposition 1. Fix γ := Z/c (0, 2/π), ℓ N0, and assume that U 0 is ∈ ∈ ≥ a measurable function on (0, ) that is form bounded with respect to Cℓ,γ, and assume that λ is sufficiently∞ small. Then | | −1 if e (0) e (λ) lim sup R3 ρℓ,Z (c x)U( x ) dx λ> 0 (11) (2ℓ + 1) n,ℓ − n,ℓ ≥ Z→∞ | | , λ −1 n lim inf RR3 ρℓ,Z (c x)U( x ) dx if λ< 0 X ≤ Z→∞ | | R where en,ℓ(λ) is the n-th eigenvalue of Cℓ,γ λU. − 4 R.L.FRANK,K.MERZ,ANDH.SIEDENTOP
Thus the limit (8) exists, if the derivative of the sum of the eigenvalues with respect to λ exists at 0, i.e., d (12) en,ℓ(λ) dλ n λ=0 X exists and – when multiplied by (2ℓ +1) – is equal to the right of (8). Put dif- ferently: the wanted limit exists,− if the Hellmann-Feynman theorem does not only hold for a single eigenvalue en,ℓ(λ) of Cℓ,γ λU but for the sum of all eigenval- ues. However, the differentiability would follow− immediately, if we were allowed to interchange the differentiation and the sum in (12), since the Hellmann-Feynman theorem is valid for each individual nondegenerate eigenvalue and yields the wanted contribution to the derivative. In turn, the validity of the interchange of these two limiting processes would follow by the Weierstraß criterion for absolute and uni- form convergence, if we had in a neighborhood of zero a λ-independent summable majorant of the moduli of the summands on the left side of (11). This, in turn is exactly the content of Lemma 1 enabling to interchange the limit λ 0 with the sum over n and concluding the proof of Theorem 1. → Before ending the section we comment on the difference to previous work: Al- ready Iantchenko et al [6, Lemma 2] proved a bound similar to (18) in the non- relativistic setting where – in contrast to the Chandrasekhar case – the hydrogenic eigenvalues are explicitly known. However, this was initially not accessible in the present context. Instead, the differentiability of the sum was shown in [2, Theo- rems 3.1 and 3.2] by an abstract argument for certain self-adjoint operators whose negative part is trace class. To prove the analogue majorant for the Chandrasekhar hydrogen operator is the new contribution of the present work yielding a substantial simplification.
3. The majorant Before giving the missing majorant, we introduce some useful notations. Set 23/2 (13) A := 2 + . π(√2 1) − For γ (0, 2/π) and t [0, 1) we set ∈ ∈ A 1+A 2 1+A 1+ t π γ (1 + t) (14) Fγ (t) := (1 t) 2 −1+t = . 2 +γ A − 1 t π 1−t γ ! π − − 1 2 t − π −γ 1 1 γ 2 Obviously F C ([ t0,t0]) with t0 := ( )/(γ + ). We set γ ∈ − π − 2 π ′ (15) M˜ := max F ([ t0,t0]). γ γ − Furthermore, we write Cγ for the optimal constant in the following inequality [4, Theorem 2.2] bounding all hydrogenic Chandrasekhar eigenvalues from below by the corresponding hydrogenic Schr¨odinger eigenvalues, i.e., (16) C e (p2/2 γ/ x ) e ( p2 +1 1 γ/ x ). γ n − | | ≤ n − − | | (Here, and sometimes also later, it is convenientp to use a slightly more general notation for eigenvalues of self-adjoint operators A which are bounded from below: RELATIVISTIC STRONG SCOTT CONJECTURE 5 we write e0(A) e1(A) for its eigenvalues below the essential spectrum counting their multiplicity.)≤ ≤ · · · Finally, we set
(17) Mγ := Cγ M˜ γ This allows us to formulate our central Lemma which allows to interchange the derivative in (12) and proves Theorem 1. We recall that we write en,ℓ(λ) for the eigenvalues of C λU (see Proposition 1). ℓ,γ − Lemma 1. Assume γ (0, 2/π) and U : R+ R+ such that the operator norm 1 1 ∈ → − 2 − 2 b := C UC is finite. Then for all λ [ t0/b,t0/b] and all ℓ,n N0 k ℓ,−γ ℓ,−γk ∈ − ∈ γ2 (18) e (λ) e (0) M b λ . | n,ℓ − n,ℓ |≤ γ | |(n + ℓ + 1)2 For the proof, we need some preparatory results. We begin with a bound on the change of Coulomb eigenvalues with the coupling constant which is the core of our argument. ′ ′ Proposition 2. For all γ,γ (0, 2/π) with γ γ and all n N0 ∈ ≤ ∈ ′ 1+A 2 A 2 ′ −1 2 −1 γ π γ (19) en( p +1 1 γ x ) en( p +1 1 γ x ) − . − − | | ≥ − − | | γ 2 γ′ π − For thep proof we will quantify thep fact that eigenfunctions live essentially in a bounded region of momentum space. Lemma 2. For all γ (0, 2/π) and all eigenfunctions ψ of p2 +1 1 γ/ x ∈ − − | | 2 2 π A 2 p−1/2 (20) ψ, p +1 1 ψ 2 ψ, 1 (p + 1) ψ . − ≤ π γ −
D p E − D E ½ Proof. Let ψ> = ½{|p|>1}ψ and ψ< = {|p|≤1}ψ. Then ψ, p2 +1 1 ψ = ψ , p2 +1 1 ψ + ψ , p2 +1 1 ψ − < − < > − > andD p E D p E D p E ψ, 1 (p2 + 1)−1/2 ψ = ψ , 1 (p2 + 1)−1/2 ψ + ψ , 1 (p2 + 1)−1/2 ψ . − < − < > − > WeD have E D E D E ψ , p2 +1 1 ψ √2 ψ , 1 (p2 + 1)−1/2 ψ , < − < ≤ < − < D p E D−1 E since sup0≤e≤1(√e +1 1)(1 1/√e + 1) = sup0≤e≤1 √e +1= √2. Moreover, by Kato’s− inequality,− 2/π ψ , p2 +1 1 ψ ψ , p ψ ψ , p γ x −1 ψ . > − > ≤ h > | | >i≤ 2/π γ > | |− | | > D E − Now usingp the eigenvalue equation for ψ we obtain ψ , p γ x −1 ψ = ψ , p γ x −1 ψ + γ ψ , x −1ψ > | |− | | > > | |− | | > | | < = ψ , E + p p2 +1+1 ψ + γ ψ , x −1ψ > | |− > | | < D E = ψ , E + p pp2 +1+1 ψ + γ ψ , x −1ψ > | |− > > | | < D E 2 ψ , 1 (p2 +p 1)−1/2 ψ + γ ψ , x −1ψ . ≤ > − > > | | < D E 6 R.L.FRANK,K.MERZ,ANDH.SIEDENTOP
In the last inequality we used E 0 (which was shown by Herbst [5, Theorem 2.2]) ≤ −1 and supe≥1(√e √e +1+1)(1 1/√e + 1) =2. Moreover, by− Hardy’s inequality− ψ , x −1ψ ψ x −1ψ 2 ψ p ψ ψ 2 + p ψ 2. h > | |
√2 2 γ ψ, p2 +1 1 ψ √2+ π ψ , 1 (p2 + 1)−1/2 ψ − ≤ √ 2 < − < ( 2 1)( π γ)! D p E − − D E 2 π √2γ 2 −1/2 (21) + 2 2+ ψ>, 1 (p + 1) ψ> γ √2 1! − π − − D E 2 π √2γ 2 −1/2 2 2+ ψ, 1 (p + 1) ψ ≤ γ √2 1! − π − − D E which gives the desired bound, since γ < 2/π.
Corollary 1. Let γ (0, 2/π) and A be the constant of the previous lemma. Then, ∈ for any normalized eigenfunction ψ of p2 +1 1 γ/ x with eigenvalue E − − | | γ p 2 A (22) ψ, ψ π +1 E . h x i≤ 2 γ | | | | π − 2 2 −1 Proof. With the abbreviation D := π A( π γ) we write the inequality in the previous lemma in the form −
p2 (23) ψ, ψ (1 + D−1) ψ, p2 +1 1 ψ . * p2 +1 + ≥ − D p E By the virial theoremp (Herbst [5, Theorem 2.4]) the left sides of (22) and (23) are equal. Thus,
p2 ψ,γ x −1ψ = (D + 1) ψ,γ x −1ψ D ψ, ψ h | | i h | | i− * p2 +1 + (D + 1) ψ,γ x −1ψ D(1 + D−1) ψ, p2p+1 1 ψ ≤ h | | i− − D E = (D + 1) ψ, p2 +1 1 γ x −1 ψp= (D + 1)E − − − | | − D E as claimed. p
We are now in position to give the RELATIVISTIC STRONG SCOTT CONJECTURE 7
Proof of Proposition 2. By the variational principle, for any n N0 the function 2 −1 ∈ κ en( p +1 1 κ x ) is Lipschitz and therefore differentiable almost ev- erywhere.7→ By perturbation− − | | theory, at every point where its derivative exists, it is p given by d e ( p2 +1 1 κ x −1)= ψ , x −1ψ , dκ n − − | | −h κ | | κi p 2 −1 where ψκ is a normalized eigenfunction of p +1 1 κ x corresponding to 2 −1 − − | | the eigenvalue en( p +1 1 κ x ).p Thus, by Corollary 1, we have for all κ (0,γ′] − − | | ∈ p 2 d 2 −1 π A −1 2 −1 en( p +1 1 κ x ) 2 +1 κ en( p +1 1 κ x ). dκ − − | | ≥ π κ − − | | p − p Thus,
d 2 −1 A +1 A (24) log en( p +1 1 κ x ) + 2 . dκ | − − | | |≤ κ π κ p − Integrating this bound we find for γ γ′ that ≤ e ( p2 +1 1 γ′ x −1) γ′ 2 γ′ log | n − − | | | (A + 1) log A log π − , e ( p2 +1 1 γ x −1) ≤ γ − 2 γ | n p − − | | | π − i.e., p A γ′ A+1 2 γ e ( p2 +1 1 γ′ x −1) e ( p2 +1 1 γ x −1) π − n n 2 ′ − − | | ≥ − − | | γ π γ p p − quod erat demonstrandum.
Eventually we can address the
Proof of Lemma 1. First let 0 < λ t0/b. Then ≤ γ 1+ bλ γ p2 +1 1 λU (1 bλ) p2 +1 1 ℓ − − r − ≥ − ℓ − − 1 bλ · r q q − and therefore for all n N0, ∈ γ 1+ bλ γ e p2 +1 1 λU (1 bλ) e p2 +1 1 . n ℓ − − r − ≥ − n ℓ − − 1 bλ · r q q − ′ ′ 1 γ 2 By Proposition 2 with γ = γ(1 + bλ)/(1 bλ) (which fulfills γ<γ π + 2 < π under our assumptions) − ≤ 1+ bλ γ e p2 +1 1 n ℓ − − 1 bλ · r q − 1+A 2 A 2 γ 1+ bλ π γ en pℓ +1 1 2 −1+bλ . ≥ − − r · 1 bλ 1 γ ! q − π − −bλ Combining the previous two inequalities shows that e (λ) F (λb)e (0) n,ℓ ≥ γ n,ℓ 8 R.L.FRANK,K.MERZ,ANDH.SIEDENTOP and therefore, if λ t0/b, ≤ λb e (λ) e (0) (F (λb) F (0)) e (0) = F ′ (t)dt e (0) n,ℓ − n,ℓ ≥ γ − γ n,ℓ γ n,ℓ Z0 γ2 M˜ b λ e (0) M b λ . ≥ γ · · · n,ℓ ≥− γ · · · (n + ℓ + 1)2 In the last inequality we used the lower bound (15). The case of negative λ is similar.
Acknowledgments The authors warmly thank Barry Simon for his initial contributions and contin- uing support and interest in the relativistic strong Scott conjecture. They also acknowledge partial support by the U.S. National Science Foundation through grants DMS-1363432 and DMS-1954995 (R.L.F.), by the Deutsche Forschungsge- meinschaft (DFG, German Research Foundation) through grant SI 348/15-1 (H.S.) and through Germany’s Excellence Strategy EXC-2111 390814868 (R.L.F., H.S.). One of us (K.M.) would like to thank the organizers of the program Density Func- tionals for Many-Particle Systems: Mathematical Theory and Physical Applications of Effective Equations, which took place at the Institute for Mathematical Sci- ences (IMS) at the National University of Singapore (NUS), for their invitation to speak, their kind hospitality, as well as for generous financial support by the Julian Schwinger foundation that made his stay possible.
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Mathematisches Institut, Ludwig-Maximilans Universitat¨ Munchen,¨ Theresienstr. 39, 80333 Munchen,¨ Germany, and Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 Munchen,¨ Germany, and Mathematics 253-37, Caltech, Pasadena, CA 91125, USA E-mail address: [email protected]
Institut fur¨ Analysis und Algebra, Carolo-Wilhelmina, Universitatsplatz¨ 2, 38106 Braunschweig E-mail address: [email protected]
Mathematisches Institut, Ludwig-Maximilans Universitat¨ Munchen,¨ Theresienstr. 39, 80333 Munchen,¨ Germany, and Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 Munchen,¨ Germany E-mail address: [email protected]