ELECTRON WAVEFUNCTIONS AND DENSITIES FOR ATOMS

arXiv:math/0005018v1 [math.AP] 2 May 2000

ELECTRON WAVEFUNCTIONS AND DENSITIES FOR ATOMS MARIA HOFFMANN-OSTENHOF, THOMAS HOFFMANN-OSTENHOF AND THOMAS ØSTERGAARD SØRENSEN Abstract. With a special ‘Ansatz’ we analyse the regularity properties of atomic electron wavefunctions and electron densities. In particular we prove an a priori estimate, supy∈B(x,R) |∇ψ(y)| ≤ C(R) supy∈B(x,2R) |ψ(y)| and obtain for the spherically averaged electron density, ρe(r), that ρe′′ (0) exists and is non-negative.

Avec un ‘Ansatz’ sp´ecial nous analysons les propriet´ees de r´egularit´e des fonctions d’ondes atomiques et des densit´es d’electron. En particulier nous prouvons une ´estimation a priori, sup |∇ψ(y)| ≤ C(R)

y∈B(x,R)

sup y∈B(x,2R)

|ψ(y)|

et obtient pour la densit´e d’electron moyenn´ee sur la sphere ρe(r), que ρe′′ (0) existe et est non-n´egative. 1. Introduction and Results

Let V be the Coulomb potential for an atom consisting of a nucleus of charge Z (fixed at the origin) and N electrons: V (x) = V (x1 , . . . , xN ) =

N X j=1

x = (x1 , . . . , xN ) ∈ R

3N

−

X Z 1 + , |xj | 1≤j 0.

(1.12)

h(x) ≤ C(R) (iv)

Z

5

ρ(y) dy + ρ(x) B(x,R)

 d2 
2 e 2 h(0) + Z ρ e (0) . ρ e (0) = dr 2 3

(1.13)

Remark 1.12. The results in (i) generalize to the case of molecules, where the continuity results for ρ and h hold in the complement of the set {R1 , . . . , RL } ⊂ R3 (see Remark 1.7). Remark 1.13. It is known that eigenfunctions obey (Kato’s) Cusp Condition (see Kato [8]), and similar properties hold for particle densities. For more recent results see Hoffmann-Ostenhof et al. [4], [5], Hoffmann-Ostenhof and Seiler [6]. In the proof of Theorem 1.11, (iv) we make use of the Cusp Condition for ρe, namely: ρe(r) − ρe(0) ρe′ (0) = lim = −Z ρe(0) and lim ρe′ (r) = ρe′ (0) (1.14) r↓0 r↓0 r and also present a proof for it. Remark 1.14. Of course our results are only first steps in a thorough investigation of qualitative properties of the one-electron density. Here are some obvious open questions: (i) Is ρ(x) > 0 for all x ∈ R3 ? We remark that this cannot be true in general, since it is false for some exited states of Hydrogen. (ii) Is ρ ∈ C ∞ (R3 \ {0}) or even C ω (R3 \ {0}) ?
dk (iii) Is ρe smooth in [0, ∞), in the sense that dr e (r) exists for r ≥ 0 kρ for all k? d ρe(r) ≤ 0 for r ≥ 0 ? This is expected to be true for (iv) Is dr groundstate densities, but not known even for the bosonic case d like Helium. Our results imply that dr ρe(r) ≤ 0 for r ≤ R0 for the bosonic case, where R0 depends on the constant C in Theorem 1.2. Note that because of (1.9) and (1.12) we have ∆ρ ≥ 0 for |x| ≥ Z/ε, and so the Maximum Principle gives that d ρe(r) < 0 for r > Z/ε. dr

Remark 1.15. In the proof of Theorem 1.11 we obtain (see Proposition 3.1): With ∇1 = ( ∂x∂1,1 , ∂x∂1,2 , ∂x∂1,3 ), the function Z Z t1 (r) = |∇1 ψ(rω, x2, . . . , xN )|2 dx2 . . . dxN dω S2

R3(N−1)

is continuous on [0, ∞).

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M. & T. HOFFMANN-OSTENHOF AND T. Ø. SØRENSEN

2. Proofs Throughout the proofs, we will denote by C generic constants. Crucial for our investigations is Corollary 8.36 in Gilbarg and Trudinger [3]. We shall make use of this result several times and for convenience we state it already here, adapted for our special case: n Proposition 2.1. Let Ω be a bounded Pn domain in R and suppose u ∈ 1,2 W (Ω) is a weak solution of ∆u + j=1 bj Dj u + W u = g in Ω, where bj , W, g ∈ L∞ (Ω). Then u ∈ C 1,α (Ω) for all α ∈ (0, 1) and for any domain Ω′ , Ω′ ⊂ Ω we have
|u|C 1,α(Ω′ ) ≤ C sup |u| + sup |g| Ω

Ω

′

for C = C(n, M, dist(Ω , ∂Ω)), with

max {1, kbj kL∞ (Ω) , kW kL∞ (Ω) , kgkL∞ (Ω) } ≤ M.

j=1,... ,n

Thereby |u|C 1,α (Ω′ ) = kukL∞ (Ω′ ) + k∇ukL∞ (Ω′ ) +

|∇u(x) − ∇u(y)| . |x − y|α x,y∈Ω′ , x6=y sup

Proof of Theorem 1.2 and Proposition 1.5. Let the function F be as in (1.6) and define the function F1 by N q X 1q X Z |xj |2 + 1 + |xj − xk |2 + 1. − F1 (x1 , . . . , xN ) = 2 4 j=1 1≤j 0, this gives −1 b − a−1 ≤ |b − a|α max{a−1−α , b−1−α }.

So, for x, y, z ∈ R3 , using max{s, t} ≤ s + t and the triangle inequality in R3 , we have  |x − z|−1 − |y − z|−1 ≤ |x − y|α |x − z|−1−α + |y − z|−1−α .

WAVEFUNCTIONS AND DENSITIES

15

In this way, by (3.4) and equivalence of norms in R3N ,  Z  |K(¯ x1 , x2 , . . . , xN )| |K(¯ x1 , x2 , . . . , xN )| (B) ≤ dx2 · · · dxN + |x1 − x2 |1+α |¯ x1 − x2 |1+α N Z Y ≤C exp(−γc0 |xj |) dxj j=3

×

Z

R3

R3



≤ C(x0 , R),

1 1 + |¯ x1 − x2 |1+α |x1 − x2 |1+α



exp(−γc0 |x2 |) dx2

since x1 , x¯1 ∈ B(x0 , R). This finishes the proof of Lemma 3.4 (a). The proof of (b) is similar to that of (a) so we omit the details. The proof of the following fact is straightforward: There exist constants C = C(γ, R) and e γ = γe(γ) such that exp(−γ|(x, . . . , xN )|) ≤ C exp(−e γ |(x0 , . . . , xN )|)

(3.6)

for all x ∈ B(x0 , R). Using this and Lemma 3.3, 3.4, we shall prove the following lemma on the regularity of the functions J1 , J2 and J3 from (3.2). Lemma 3.5. Let J1 , J2 and J3 be as in (3.2). Then α (i) J2 , J3 ∈ Cloc (R3 ) for all α ∈ (0, 1). α (ii) J1 ∈ Cloc (R3 \ {0}) for all α ∈ (0, 1).

Herefrom follow the regularity properties of the function h stated in Proposition 3.1. Proof of Lemma 3.5 (i). Firstly, by Theorem 1.2 and Remark 1.8, |ψ(x)| , |∇ψ(x)| ≤ C exp(−γ|x|) , x ∈ R3N ,

(3.7)

which gives (3.4) for K = ψ 2 . Next we verify that for G = ψ 2 , (3.3) is fulfilled. Then Lemma 3.4 can be applied with K = ψ 2 , and the H¨older-continuity of J2 and J3 follows. Given x0 ∈ R3 , R > 0, α ∈ (0, 1), and x, y ∈ B(x0 , R). Using that (see e. g. Mal´y and Ziemer [11, Theorem 1.41]) (here, (x, x2 , . . . , xN ) = (x, x′ ), x′ ∈ R3(N −1) ) Z 1  ∂ 2 2 ′ 2 ′ ψ (sx + (1 − s)y, x′) ds ψ (x, x ) − ψ (y, x ) = 0 ∂s Z 1   = ∇1 (ψ 2 )(sx + (1 − s)y, x′) · (x − y) ds (3.8) 0

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M. & T. HOFFMANN-OSTENHOF AND T. Ø. SØRENSEN

and that sx + (1 − s)y ∈ B(x0 , R) for all s ∈ [0, 1] we get, with (3.7) and (3.6), |ψ 2 (x, x′ ) − ψ 2 (y, x′ )| |x − y|α Z 1 1−α ≤ 2|x − y| |∇1 ψ(sx + (1 − s)y, x′ )| · |ψ(sx + (1 − s)y, x′ )| ds 0 Z 1 C exp(−2γ|(sx + (1 − s)y, x′ )|) ds ≤ 2(2R)1−α 0

≤ C exp(−e γ |(x0 , . . . , xN )|),

(3.9)

so (3.3) follows for α ∈ (0, 1). This proves (i) of Lemma 3.5. To prove (ii), we write ψ as in the proof of Theorem 1.2: ψ = eF −F1 ψ1 , with F and F1 as in (1.6) and (2.1). Then Z Z Z 2 ′ 2 2 ′ J1 (x) = |∇ψ| dx = |∇F | ψ dx + |∇F1 |2 ψ 2 dx′ Z Z
2 ′ −2 ∇F · ∇F1 ψ dx + e2(F −F1 ) |∇ψ1 |2 dx′ Z Z
2(F −F1 )
′ +2 ∇F · ∇ψ1 e ψ1 dx − 2 ∇F1 · ∇ψ1 e2(F −F1 ) ψ1 dx′ ≡ I1 (x) + I2 (x) + I3 (x) + I4 (x) + I5 (x) + I6 (x).

(3.10)

Using the idea from (3.8) twice (on |∇F1 |2 and ψ 2 , respectively), the estimates (2.3), (3.7), (3.9), and (3.6), we have, with x0 ∈ R3 , R > 0, α ∈ (0, 1), and x, y ∈ B(x0 , R): |∇F1 |2 ψ 2 (x, x′ ) − |∇F1 |2 ψ 2 (y, x′) ≤ |x − y|α |∇F1 (x, x′ )|2 − |∇F1 (y, x′ )|2 |ψ(x, x′ )|2 |x − y|α 2 ′ 2 ′ ′ 2 ψ (x, x ) − ψ (y, x x) + |∇F1 (y, x )| |x − y|α


≤ C |x − y|1−α ∇ |∇F1 |2 ∞ exp(−2γ|(x, x2 , . . . , xN )|)

e − y|1−α |∇F1 |2 exp(−e + 2C|x γ |(x0 , . . . , xN )|) ∞

≤ C¯ exp(−¯ γ (x0 , . . . , xN )|).

By Lemma 3.3, with G = |∇F1 |2 ψ 2 (x1 , · · · , xN ), this implies that I2 ∈ α Cloc (R3 ). Using the same ingredients, writing ∇ψ1 = ∇(eF1 −F ψ), gives (3.3) and (3.4) with G = K = e2(F −F1 ) |∇ψ1 |2

WAVEFUNCTIONS AND DENSITIES

17

and
G = K = ∇F1 · ∇ψ1 e2(F −F1 ) ψ1 ,

α and so by Lemma 3.3, I4 , I6 ∈ Cloc (R3 ). The remaining terms are those involving the function F , namely I1 , I3 and I5 . Note that

Z  x1 xN  ∇F (x1 , . . . , xN ) = − ,... , 2 |x1 | |xN | ! N N N −1 X xj − xk X 1 X x1 − xk xN − xk + ,... , ,... , 4 k=2 |x1 − xk | |x |xN − xk | j − xk | k=1,k6=j k=1

(3.11)

and so |∇F (x1 , . . . , xN )|2 =

NZ 2 Z − 4 8 1 + 16

N X

xj xj − xk · |xj | |xj − xk | j,k=1,k6=j

N X

j,k,l=1,k6=j,l6=j

xj − xl xj − xk · . |xj − xk | |xj − xl |

In this way, NZ 2 I1 (x1 ) = 4 Z − 8

Z

ψ 2 dx2 · · · dxN N X

j,k=1,k6=j

Z

xj − xk 2 xj · ψ dx2 · · · dxN |xj | |xj − xk |

N X 1 xj − xk xj − xl 2 + · ψ dx2 · · · dxN 16 j,k,l=1,k6=j,l6=j |xj − xk | |xj − xl |

Z NZ 2 ρ(x1 ) − = 4 8

N X

N X 1 κj,k (x1 ) + νj,k,l (x1 ). (3.12) 16 j,k,l=1,k6=j,l6=j j,k=1,k6=j

Note that νj,k,l = νj,l,k . Using the ideas above, Lemma 3.3 implies that the following functions from (3.12) (with the mentioned choices of G satisfying (3.3)) are

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M. & T. HOFFMANN-OSTENHOF AND T. Ø. SØRENSEN

α all in Cloc (R3 ):

ρ: κj,k ,

j, k 6= 1, j 6= k : νj,k,k , j 6= k :

νj,k,l,

j, k, l 6= 1, l 6= j 6= k :

G = ψ2 , xj xj − xk 2 G= · ψ , |xj | |xj − xk | xj − xk 2 xj − xk · ψ = ψ2, G= |xj − xk | |xj − xk | xj − xl 2 xj − xk · ψ . G= |xj − xk | |xj − xl |

Likewise, Lemma 3.4 implies (with the mentioned choices of G = K satisfying (3.3) and (3.4)) that the following functions from (3.12) are α all in Cloc (R3 ): κj,1 , νj,1,l ,

j 6= 1 :

j, l 6= 1, j 6= l :

xj · (xj − x1 ) 2 ψ , |xj | (xj − x1 ) · (xj − xl ) 2 ψ , G=K= |xj − xl |

G=K=

From the decomposition of I1 in (3.12) we are left with Z x1 x1 − xk 2 ′ κ1,k (x1 ) = · ψ dx , k = 2, . . . , N, |x1 | |x1 − xk |

(3.13)

and ν1,k,l (x1 ) =

Note that Z

Z

x1 − xl 2 ′ x1 − xk · ψ dx |x1 − xk | |x1 − xl |

,

k, l ∈ {2, . . . , N}, k 6= l. (3.14)

x1 x1 − xk 2 · ψ dx2 · · · dxN |x1 | |x1 − xk | Z
1 1 = x1 · (x1 − xk ) ψ 2 dx2 · · · dxN . |x1 | |x1 − xk |

α The function 1/|x1 | is smooth for x1 6= 0 and therefore in Cloc (R3 \{0}). The function x1 ·(x1 −xk ) ψ 2 satisfies (3.3) and (3.4) (by the same ideas as above), so Lemma 3.4 (a) implies that the function Z
1 x1 · (x1 − xk ) ψ 2 dx2 · · · dxN |x1 − xk |

α α is in Cloc (R3 ). The functions in (3.13) are therefore in Cloc (R3 \ {0}). α As for the functions in (3.14), these are all in Cloc (R3 ), which can be seen by applying the previous ideas, in particular Lemma 3.5, (3.6), (3.7) and (3.9). α This proves that I1 ∈ Cloc (R3 \ {0}).

WAVEFUNCTIONS AND DENSITIES

19

As for I3 (see (3.10) and (3.11)), with ∇ = (∇1 , . . . , ∇N ), N Z   X xj I3 (x) = Z · ∇j F1 ψ 2 dx2 · · · dxN |xj | j=1

Z  N  xj − xk 1 X − · ∇j F1 ψ 2 dx2 · · · dxN . 2 j,k=1,j6=k |xj − xk |

(3.15)

α The terms in the first sum with j 6= 1 are in Cloc (R3 ), due to Lemma 3.4 (b), with G = K = (xj ·∇j F1 ) ψ 2 satisfying (3.3) and (3.4). (To see this, use the previous ideas; to apply the idea from (3.8) to ∇j F1 we use that α 3 F1 is smooth). The terms in the second sum in (3.15) are all in Cloc (R
2), due to Lemma 3.4 (a), applied with G = K = (xj − xk ) · ∇j F1 ψ . α The term with j = 1 is in Cloc (R3 \ {0}). This can be seen by following the ideas in the proof of the regularity properties of the function in (3.13), now using Lemma 3.3 with G = (x1 · ∇1 F1 )ψ 2 . The statements and proofs are similar for N Z   X xj I5 (x) = −Z · ∇j ψ1 e2(F −F1 ) ψ1 dx2 · · · dxN |xj | j=1

Z  N  xj − xk 1 X · ∇j ψ1 e2(F −F1 ) ψ1 dx2 · · · dxN . + 2 j,k=1,j6=k |xj − xk |

(3.16)

That is, the functions in the first sum in (3.16) with j ≥ 2 and those α in the second sum are all in Cloc (R3 ), whereas the function in the first α sum with j = 1 is only in Cloc (R3 \ {0}). To prove this we use the inequality (with x = (x1 , . . . , xN )): |ψ1 |C 1,α (B(x,R/2)) ≤ C

sup y∈(B(x,R))

|ψ1 (y)| ≤ C exp(−γ|(x1 , . . . , xN )|).

This inequality follows from (2.3), (2.5) and (3.7) (remember that ψ1 = eF1 −F ψ). α This proves that I5 ∈ Cloc (R3 \ {0}), and so finishes the proof that α 3 J1 ∈ Cloc (R \ {0}). (See (3.10)). This proves (ii) and therefore Lemma 3.5. α That e h ∈ Cloc ((0, ∞)) is a consequence of the foregoing and of the following proposition: α Proposition 3.6. Assume f ∈ Cloc (R3 \ {0}), α ∈ (0, 1). Then fe ∈ α Cloc ((0, ∞)), where Z fe(r) = f (rω) dω. S2

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M. & T. HOFFMANN-OSTENHOF AND T. Ø. SØRENSEN

Proof. Let r ∈ (0, ∞). For all x0 ∈ A = {x ∈ R3 | |x| = r}, choose R = R(x0 ) and C = C(x0 ) such that |f (x) − f (y)| ≤ C(x0 ). |x − y|α x,y∈B(x0 ,R(x0 )) sup

(3.17)

α This is possible, since f ∈ Cloc (R3 \ {0}). Then [ A⊂ B(x0 , R(x0 )). x0 ∈A

Using compactness of A, choose x1 , . . . , xm ∈ A such that A⊂

m [

B(xj , R(xj )).

j=1

Choose ǫ ∈ (0, r) such that 3

{y ∈ R | r − ǫ < |y| < r + ǫ} ⊂

m [

B(xj , R(xj )).

j=1

Then, for all s, t ∈ (r − ǫ, r + ǫ) and all ω ∈ S2 there exists j ∈ {1, . . . , m} such that sω, tω ∈ B(xj , R(xj )) and therefore by (3.17), |f (sω) − f (tω)| |f (sω) − f (tω)| = ≤ C(xj ). α |s − t| |sω − tω|α

So with C = max{C(x1 ), . . . , C(xm )},

|f (sω) − f (tω)| ≤ C, for all s, t ∈ (r − ǫ, r + ǫ) and all ω ∈ S2 . |s − t|α

This implies that

R | S2 (f (sω) − f (tω)) dω| |fe(s) − fe(t)| = |s − t|α |s − t|α Z |f (sω) − f (tω)| ≤ dω ≤ C, for all s, t ∈ (r − ǫ, r + ǫ). |sω − tω|α S2

α This proves that fe ∈ Cloc ((0, ∞)).

To prove that e h ∈ C 0 ([0, ∞)), we apply the following:

α Proposition 3.7. Assume f ∈ Cloc (R3 ). Then fe ∈ C 0 ([0, ∞)), where Z e f (r) = f (rω) dω. S2

α Proof. The function f is continuous in R3 , since it is in Cloc (R3 ). Let r ∈ [0, ∞). Then

lim f (sω) = f (rω)

s→r

for all ω ∈ S2 .

WAVEFUNCTIONS AND DENSITIES

21

Using the supremum of f on a sufficiently large compact set in R3 as a dominant, Lebesque’s Dominated Convergence Theorem gives us that Z Z f (sω) dω = f (rω) dω. lim s→r

S2

S2

Therefore f ∈ C 0 ([0, ∞)).

α Recall the proof of the fact that h ∈ Cloc (R3 \ {0}). In fact, the only terms in the decomposition of h (see (3.1), (3.2), (3.10), and (3.11)) α α that are only in Cloc (R3 \ {0}) and not in Cloc (R3 ) are the functions Z x1 − xk 2 x1 · ψ dx2 · · · dxN , k = 2, . . . , N, |x1 | |x1 − xk | Z   x1 · ∇1 F1 ψ 2 dx2 · · · dxN , |x | Z  1  x1 · ∇1 ψ1 e2(F −F1 ) ψ1 dx2 · · · dxN . (3.18) |x1 |

Comparing (3.10), (3.13), (3.15) and (3.16), it can be seen that all the terms in (3.18) stem from the function J1 , namely from I1 , I3 , and I5 . α All other terms in the decomposition of h are in Cloc (R3 ). When inte2 grating them over S , we get something continuous in [0, ∞), according to Proposition 3.7 above. For the terms in (3.18) we note that they are all of the form Z Z x1 x1 ′ ′ · K(x1 , x ) dx = · K(x1 , x′ ) dx′ . (3.19) |x1 | |x1 | In each case, we have




L(x1 ) = L1 (x1 ), L2 (x1 ), L3 (x1 ) =

Z

α K(x1 , x′ ) dx′ , Lj ∈ Cloc (R3 ).

(3.20)

To see this, apply Lemma 3.3 to each of the coordinate functions Lj , j = 1, 2, 3. The integrands are easily seen to satisfy (3.3) in each case, by the previous ideas. To get continuity in [0, ∞) of the functions in (3.18) we use (3.19) and (3.20), and the following lemma: α Proposition 3.8. Assume f = (f1 , f2 , f3 ), fj ∈ Cloc (R3 ). Then f¯ ∈ 0 C ([0, ∞)), where Z
¯ f (r) = ω · f(rω) dω. S2

Proof. The same as for Proposition 3.7, noting that for all r ∈ [0, ∞) and fixed ω ∈ S2 : lim ω · f(sω) = ω · f(rω).

s→r

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M. & T. HOFFMANN-OSTENHOF AND T. Ø. SØRENSEN

This holds even in the case r = 0, for which Z Z ω · f(sω) dω = ω · f(0) dω = 0. lim s↓0

S2

S2

This proves that the functions in (3.18) are in C 0 ([0, ∞)). Therefore e h ∈ C 0 ([0, ∞)), which finishes the proof of Proposition 3.1.

Acknowledgement: The authors wish to thank G. Friesecke and S. Fournais for stimulating discussions. References

[1] R. Ahlrichs, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and J.D. Morgan, III, Bounds on the decay of electron densities with screening, Phys. Rev. A (3) 23 (1981), 2106–2117. [2] H.L. Cycon, R.G. Froese, W. Kirsch, and B. Simon, Schr¨ odinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. [3] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, second ed., Springer-Verlag, Berlin, 1983. [4] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and H. Stremnitzer, Local properties of Coulombic wave functions, Comm. Math. Phys. 163 (1994), 185– 215. [5] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and W. Thirring, Simple bounds to the atomic one-electron density at the nucleus and to expectation values of one-electron operators, J. Phys. B 11 (1978), L571–L575. [6] M. Hoffmann-Ostenhof and R. Seiler, Cusp conditions for eigenfunctions of n-electron systems, Phys. Rev. A 23 (1981), 21–23. ¨ [7] E. Hopf, Uber den funktionalen, insbesondere den analytischen Charakter der L¨ osungen elliptischer Differentialgleichungen zweiter Ordnung, Math. Z. 34 (1931), 194–233. [8] T. Kato, On the eigenfunctions of many-particle systems in quantum mechanics, Comm. Pure Appl. Math. 10 (1957), 151–177. [9] E.H. Lieb, The Stability of Matter: From Atoms to Stars. Selecta of Elliot H. Lieb, second ed., Springer - Verlag, Berlin/Heidelberg, 1997. [10] E.H. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, RI, 1997. [11] Jan Mal´ y and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, American Mathematical Society, Providence, RI, 1997. [12] M. Reed and B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press, New York, 1975. [13] , Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, New York, 1978. [14] B. Simon, Schr¨ odinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447–526. [15] G.M. Zhislin and A.G. Sigalov, On the spectrum of the energy operator for atoms with fixed nuclei on subspaces corresponding to irreducible representations of a group of permutations, Am. Math. Soc., Transl., II. Ser. 91 (1970), 263–295. ¨r Mathematik, Strudlhofgasse 4, (M. Hoffmann-Ostenhof) Institut fu ¨t Wien, A-1090 Vienna, Austria Universita

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¨r Theoretische Chemie, Wa ¨hringer(T. Hoffmann-Ostenhof) Institut fu ¨t Wien, A-1090 Vienna, Austria strasse 17, Universita ¨ dinger Interna(T. Hoffmann-Ostenhof, T. Ø. Sørensen) The Erwin Schro tional Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria E-mail address, M. Hoffmann-Ostenhof: [email protected] E-mail address, T. Hoffmann-Ostenhof: [email protected] E-mail address, T. Ø. Sørensen: [email protected]