Tài liệu Đề tài " The ionization conjecture in Hartree-Fock theory " pptx

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512 JAN PHILIP SOLOVEJ
ϕ
HF
(x):=Z|x|
−1
−
ρ
HF
∗|x|
−1
= Z|x|
−1
−

ρ
HF
(y)|x −y|
−1
dy(1)
ϕ
TF
(x):=Z|x|
−1
−
ρ
TF
∗|x|
−1
= Z|x|
−1
−

ρ
TF
(y)|x −y|
−1
dy(2)
and for all R ≥ 0 the screened nuclear potentials at radius R
Φ
HF
R
(x):=Z|x|
−1
−

|y|<R
ρ
HF
(y)|x −y|
−1
dy(3)
Φ
TF
R
(x):=Z|x|
−1
−

|y|<R
ρ
TF
(y)|x −y|
−1
dy.(4)
This is the potential from the nuclear charge Z screened by the electrons in the
region {x : |x| <R}. The screened nuclear potential will be very important in
the technical proofs in Sections10–13.
Definition 1.2. (Radius). Let again
ρ
HF
and
ρ
TF
be the densities of atomic
ground states in the HF and TF models respectively. We define the radius
R
Z,N
(ν)tothe ν last electrons by

|x|≥R
TF
Z,N
(ν)
ρ
TF
(x) dx = ν,

|x|≥R
HF
Z,N
(ν)
ρ
HF
(x) dx = ν.
The functions ϕ
TF
and
ρ
TF
are the unique solutions to the set of equations
∆ϕ
TF
(x)=4π
ρ
TF
(x) − 4πZδ(x)(5)
ρ
TF
(x)=2
3/2
(3π
2
)
−1
[ϕ
TF
(x) − µ
TF
]
3/2
+
(6)

ρ
TF
= N.(7)
Here µ
TF
is a nonnegative parameter called the chemical potential, which is
also uniquely determined from the equations. We have used the notation
[t]
+
= max{t, 0} for all t ∈ . The equations (5–7) only have solutions when
N ≤ Z.ForN>Zwe shall let ϕ
TF
and
ρ
TF
refer to the solutions for N = Z,
the neutral case. Instead of fixing N and determining µ
TF
(the ‘canonical’ pic-
ture) one could fix µ
TF
and determine N (the ‘grand canonical’ picture). The
equation (5) is essentially equivalent to (2) and expresses the fact that ϕ
TF
is
the mean field potential generated by the positive charge Z and the negative
charge distribution −
ρ
TF
. The equations (6–7) state that
ρ
TF
is given by the
semiclassical expression for the density of an electron gas of N electrons in the
exterior potential ϕ
TF
.For a discussion of semiclassics we refer the reader to
Section 8.
Remark 1.3. The total energy of the atom in Thomas-Fermi theory is
3
10
(3π
2
)
2/3

ρ
TF
(x)
5/3
dx − Z

ρ
TF
(x)|x|
−1
dx(8)
+
1
2

ρ
TF
(x)|x − y|
−1
ρ
TF
(y)dx dy ≥−e
0
Z
7/3
THE IONIZATION CONJECTURE IN HARTREE-FOCK THEORY 513
where e
0
is the total binding energy of a neutral TF atom of unit nuclear
charge. Numerically [10],
(9) e
0
= 2(3π
2
)
−2/3
· 3.67874 = 0.7687.
Foraneutral atom, where N = Z, the above inequality is an equality. The
inequality states that in Thomas-Fermi theory the energy is smallest for a
neutral atom.
We can now state two of the main results in this paper.
Theorem 1.4 (Potential estimate). For al l Z ≥ 1 and all integers N
with N ≥ Z for which there exist Hartree-Fock ground states with

ρ
HF
= N
we have
(10) |ϕ
HF
(x) − ϕ
TF
(x)|≤A
ϕ
|x|
−4+ε
0
+ A
1
,
where A
ϕ
,A
1
,ε
0
> 0 are universal constants.
This theorem is proved in Section 13 on page 535. The significance of the
power |x|
−4
is that for N ≥ Z we have lim
Z→∞
ϕ
TF
(x)=3
4
2
−3
π
2
|x|
−4
. The
existence of this limit known as the Sommerfeld asymptotic law [27] follows
from Theorem 2.10 in [10], but we shall also prove it in Theorems 5.2 and 5.4
below.
Note that the bound in Theorem 1.4 is uniform in N and Z.
The second main theorem is the universal bound on the atomic radius
mentioned in the beginning of the introduction. In fact, not only do we prove
uniform bounds but we also establish a certain exact asymptotic formula for
the radius of an “infinite atom”.
Theorem 1.5. Both lim inf
Z→∞
R
HF
Z,Z
(ν) and lim sup
Z→∞
R
HF
Z,Z
(ν) are bounded
and have the asymptotic behavior
2
−1/3
3
4/3
π
2/3
ν
−1/3
+ o(ν
−1/3
)
as ν →∞.
The proof of this theorem can be found in Section 13 on page 535. The
universal bound on the maximal ionization is given in Theorem 3.6. The proof
is given in Section 13 on page 534. A universal bound on the ionization energy
(the energy it takes to remove one electron) is formulated in Theorem 3.8.
The proof is given in Section 13 on page 537. Theorems 3.6 and 3.8 are
as important as Theorems 1.4 and 1.5. We have deferred the statements of
Theorems 3.6 and 3.8 in order not to have to make too many definitions here
in the introduction.
One of the main ideas in the paper is to use the strong universal behav-
ior of the TF theory reflected in the Sommerfeld asymptotics. If we com-
bine (5) and (6) we see that for µ
TF
=0the potential satisfies the equation
514 JAN PHILIP SOLOVEJ
∆ϕ
TF
(x)=2
7/2
(3π)
−1
[ϕ
TF
]
3/2
+
(x) for x =0.Itturns out that the singularity
at x =0of any solution to this equation is either of weak type ∼ Z|x|
−1
for
some constant Z or of strong type ∼ 3
4
2
−3
π
2
|x|
−4
(see [30] for a discussion
of singularities for differential equations of similar type). The surprising fact,
contained in Theorem 1.4, is that the same type of universal behavior holds
also for the much more complicated HF potential. We prove this by comparing
with appropriately modified TF systems on different scales, using the fact that
the modifications do not affect the universal behavior. A direct comparison
works only in a short range of scales. This is however enough to use an iter-
ative renormalization argument to bootstrap the comparison to essentially all
scales.
The paper is organized as follows. In Section 2 we fix our notational
conventions and give some basic prerequisites. In Section 3 we discuss Hartree-
Fock theory. In Sections 4 and 5 we discuss Thomas-Fermi theory. In particular
we show that the TF model, indeed, has the universal behavior for large Z that
we want to establish for the HF model. In the TF model the universality can
be expressed very precisely through the Sommerfeld asymptotics.
In Section 6 we begin the more technical work. We show in this section
that the HF atom in the region {x : |x| >R} is determined to a good approx-
imation, in terms of energy, from knowledge of the screened nuclear potential
Φ
HF
R
.Itisthis crucial step in the whole argument that I do not know how to
generalize to the Schr¨odinger model or even to the case of molecules in HF
theory.
For the outermost region of the atom one cannot use the energy to control
the density. In fact, changing the density of the atom far from the nucleus will
not affect the energy very much. Far away from the nucleus one must use the
exact energy minimizing property of the ground state, i.e., that it satisfies a
variational equation. This is done in Section 7 to estimate the L
1
-norm of the
density in a region of the form {x : |x| >R}.
In Section 8 we establish the semiclassical estimates that allow one to
compare the HF model with the TF model. To be more precise, there is no
semiclassical parameter in our setup, but we derive bounds that in a semiclas-
sical limit would be asymptotically exact.
It turns out to be useful to use the electrostatic energy (or rather its square
root) as a norm in which to control the difference between the densities in TF
and HF theory. The properties of this norm, which we call the Coulomb norm,
are discussed in Section 9. Sections 4–9 can be read almost independently.
In Section 10 we state and prove the main technical tool in the work. It is
a comparison of the screened nuclear potentials in HF and TF theory. Using a
comparison between the screened nuclear potentials at radius R one may use
the result of the separation of the outside from the inside given in Section 6 to
THE IONIZATION CONJECTURE IN HARTREE-FOCK THEORY 515
get good control on the outside region {x : |x| >R}. Using an iterative scheme
one establishes the main estimate for all R. The two main technical lemmas
are proved in Section 11 and Section 12 respectively.
Finally the main theorems are proved in Section 13.
The main results of this paper were announced in [26] and a sketch of the
proof was given there. The reader may find it useful to read this sketch as a
summary of the proof.
2. Notational conventions and basic prerequisites
We shall throughout the paper use the definitions
B(r):=

y ∈
3
: |y|≤r

,(11)
B(x, r):=

y ∈
3
: |y − x|≤r

,(12)
A(r
1
,r
2
):=

x ∈
3
: r
1
≤|x|≤r
2

.(13)
For any r>0weshall denote by
χ
r
the characteristic function of the ball
B(r) and by
χ
+
r
=1−
χ
r
.Weshall as in the introduction use the notation
[t]
±
=(t)
±
:= max{±t, 0}.
Our convention for the Fourier transform is
(14)
ˆ
f(p):=(2π)
−3/2

e
ipx
f(x) dx.
Then
(15)

f ∗ g =(2π)
3/2
ˆ
f ˆg, f
2
= 
ˆ
f
2
, |
ˆ
f(p)|≤(2π)
−3/2
f
1
and
(16)

f(x)|x −y|
−1
g(y)dx dy = 2(2π)

ˆ
f(p)
ˆg(p)|p|
−2
dp.
Definition 2.1. (Density matrix). Here we shall use the definition that a
density matrix,onaHilbert space H,isapositive trace class operator satisfying
the operator inequality 0 ≤ γ ≤ I. When H is either L
2
(
3
)orL
2
(
3
;
2
)we
write γ(x, y) for the integral kernel for γ.Itis2×2 matrix valued in the case
L
2
(
3
;
2
). We define the density 0 ≤
ρ
γ
∈ L
1
(
3
) corresponding to γ by
(17)
ρ
γ
:=

j
ν
j
|u
j
(x)|
2
,
where ν
j
and u
j
are the eigenvalues and corresponding eigenfunctions of γ.
Then

ρ
γ
=Tr[γ].
Remark 2.2. Whenever γ is a density matrix with eigenfunctions u
j
and
corresponding eigenvalues ν
j
on either L
2
(
3
)orL
2
(
3
;
2
)weshall write
(18) Tr [−∆γ]:=

j
ν
j

|∇u
j
(x)|
2
dx.
516 JAN PHILIP SOLOVEJ
If we allow the value +∞ then the right side is defined for all density matrices.
The expression −∆γ may of course define a trace class operator for some γ,
i.e., if the eigenfunctions u
j
are in the Sobolev space H
2
and the right side
above is finite. In this case the left side is well defined and is equal to the right
side. On the other hand, the right side may be finite even though −∆γ does
not even define a bounded operator, i.e., if an eigenfunction is in H
1
, but not
in H
2
. Then the sum on the right is really
Tr

(−∆)
1/2
γ(−∆)
1/2

=Tr[∇·γ∇] .
It is therefore easy to see that (18) holds not only for the spectral decompo-
sition, but more generally, whenever γ can be written as γf =

j
ν
j
(u
j
,f)u
j
,
with 0 ≤ ν
j
(the u
j
need not be orthonormal). The same is also true for the
expression (17) for the density.
Proposition 2.3 (The radius of an infinite neutral HF atom). The map
γ → Tr[−∆γ] as defined above on all density matrices is affine and weakly
lower semicontinuous.
Proof. Choose a basis f
1
,f
2
, for L
2
consisting of functions from H
1
.
Then
Tr[−∆γ]=

m
(∇f
m
,γ∇f
m
).
The affinity is trivial and the lower semicontinuity follows from Fatou’s lemma.
We are of course abusing notation when we define Tr[−∆γ] for all density
matrices. This is, however, very convenient and should hopefully not cause
any confusion.
If V is a positive measurable function, we always identify V with a mul-
tiplication operator on L
2
.IfV
ρ
γ
∈ L
1
(
3
)weabuse notation and write
Tr [Vγ]:=

V
ρ
γ
.
As before if Vγ happens to be trace class then the left side is well defined
and finite and is equal to the right side. Otherwise, we really have

V
ρ
γ
=
Tr

[V ]
1/2
+
γ[V ]
1/2
+

− Tr

[V ]
1/2
−
γ[V ]
1/2
−

.
Lemma 2.4 (The IMS formulas). If u is in the Sobolev space H
1
(
3
;
2
)
or H
1
(
3
) and if Ξ ∈ C
1
(
3
) is real, bounded, and has bounded derivative
then
1
(19) Re

∇

Ξ
2
u
∗

·∇u =

|∇(Ξu)|
2
−

|∇Ξ|
2
|u|
2
.
1
We denote by u
∗
the complex conjugate of u.Inthe case when u takes values in
2
this refers
to the complex conjugate matrix.
THE IONIZATION CONJECTURE IN HARTREE-FOCK THEORY 517
If γ is a density matrix on L
2
(
3
;
2
) or L
2
(
3
) and if Ξ
1
, ,Ξ
m
∈ C
1
(
3
)
are real, bounded, have bounded derivatives, and satisfy Ξ
2
1
+ +Ξ
2
m
=1then
Tr [−∆γ]=Tr[−∆(Ξ
1
γΞ
1
)] − Tr

(∇Ξ
1
)
2
γ

+ (20)
+Tr[−∆(Ξ
m
γΞ
m
)] − Tr

(∇Ξ
m
)
2
γ

.
Note that Ξ
j
γΞ
j
again defines a density matrix (where we identified Ξ
j
with a
multiplication operator).
Proof. The identity (19) follows from a simple computation. If we sum
this identity and use Ξ
2
1
+ +Ξ
2
m
=1we obtain

|∇u|
2
=

|∇(Ξ
1
u)|
2
−

|∇Ξ
1
|
2
|u|
2
+ +

|∇(Ξ
m
u)|
2
−

|∇Ξ
m
|
2
|u|
2
.
If we allow the value +∞ this identity holds for all functions u in L
2
.Thus
(20) is a simple consequence of the definition (18).
Theorem 2.5 (Lieb-Thirring inequality). We have the Lieb-Thirring
inequality
(21) Tr

−
1
2
∆γ

≥ K
1

ρ
5/3
γ
,
where K
1
:= 20.49. Equivalently, If V ∈ L
5/2
(
3
) and if γ is any density
matrix such that Tr[−∆γ] < ∞ we have
(22) Tr

−
1
2
∆γ

− Tr [Vγ] ≥−L
1

[V ]
5/2
+
,
where L
1
:=
2
5

3
5K
1

2/3
=0.038.
The original proofs of these inequalities can be found in [18]. The con-
stants here are taken from [7]. From the min-max principle it is clear that the
right side of (22) is in fact a lower bound on the sum of the negative eigenvalues
of the operator −
1
2
∆ − V .
Theorem 2.6 (Cwikel-Lieb-Rozenblum inequality). If V ∈ L
3/2
(
3
)
then the number of nonpositive eigenvalues of −
1
2
∆ − V , i.e.,
Tr

χ
(−∞,0]

−
1
2
∆ − V

,
where χ
(−∞,0]
is the characteristic function of the interval (−∞, 0], satisfies
the bound
(23) Tr

χ
(−∞,0]

−
1
2
∆ − V

≤ L
0

[V ]
3/2
+
,
where L
0
:= 2
3/2
0.1156 = 0.3270.
518 JAN PHILIP SOLOVEJ
The original (independent) proofs can be found in Cwikel [4], Rozen-
blum [19], and Lieb [9]. The constant is from Lieb [9].
3. Hartree-Fock theory
In Hartree-Fock theory, as opposed to Schr¨odinger theory, one does not
consider the full N-body Hilbert space

N
L
2
(
3
;
2
). One rather restricts
attention to the pure wedge products (Slater determinants)
(24) Ψ = (N!)
−1/2
u
1
∧ ∧ u
N
,
where u
1
, ,u
N
∈ L
2
(
3
;
2
). Then one minimizes the energy expectation
(Ψ,H
N,Z
Ψ)
(Ψ, Ψ)
of the Hamiltonian
(25) H
N,Z
:=
N

i=1

−
1
2
∆ −
Z
|x|

+

1≤i<j≤N
1
|x
i
− x
j
|
over wave functions Ψ of the form (24) only.
If γ is the projection onto the N-dimensional space spanned by the func-
tions u
1
, ,u
N
, the energy depends only on γ.Infact,
(Ψ,H
N,Z
Ψ)
(Ψ, Ψ)
= E
HF
(γ).
Here we have defined the Hartree-Fock energy functional
E
HF
(γ) :=Tr

−
1
2
∆ − Z|x|
−1

γ

+ D(γ) −EX (γ)(26)
=Tr

−
1
2
∆γ

−

Z|x|
−1
ρ
γ
(x) dx + D(γ) −EX (γ),
where we have introduced the direct Coulomb energy, defined in terms of the
Coulomb inner product D (see also (79) below), by
(27) D(γ):=D(
ρ
γ
,
ρ
γ
)=
1
2

ρ
γ
(x)|x − y|
−1
ρ
γ
(y)dx dy
and the exchange Coulomb energy
(28) E
X (γ):=
1
2

Tr
2

|γ(x, y)|
2

|x − y|
−1
dx dy.
Definition 3.1. (The Hartree-Fock ground state). Let Z>0beareal
number and N ≥ 0beaninteger. The Hartree-Fock ground state energy is
E
HF
(N,Z):=inf

E
HF
(γ):γ
∗
= γ, γ = γ
2
, Tr[γ]=N

.
If a minimizer
γ
HF
exists we say that the atom has an HF ground state described
by
γ
HF
.Inparticular, its density is
ρ
HF
(x)=
ρ
γ
HF
(x).
THE IONIZATION CONJECTURE IN HARTREE-FOCK THEORY 519
Theorem 3.2 (Bound on the Hartree-Fock energy). For Z>0 and any
integer N>0 we have
E
HF
(N,Z) ≥−3(4πL
1
)
2/3
Z
2
N
1/3
,
where L
1
is the constant in the Lieb-Thirring inequality (22).
Proof. Let γ be an N dimensional projection. Since the last term in H
N,Z
is positive we see that E
HF
(γ) ≥ Tr

−
1
2
∆ − Z|x|
−1

γ

.Itthe follows from
the Lieb-Thirring inequality (22) that for all R>0wehave
E
HF
(γ) ≥−L
1

|x|<R
Z
5/2
|x|
−5/2
dx − ZNR
−1
.
The estimate in the theorem follows by evaluating the integral and choosing
the optimal value for R.
Remark 3.3. The function N → E
HF
(N,Z)isnonincreasing. This can
be seen fairly easily by constructing a trial N + 1-dimensional projection from
any N-dimensional projection by adding an extra dimension corresponding to
a function u concentrated far from the origin and with very small kinetic energy

|∇u|
2
. This trial projection can be constructed such that it has an energy
arbitrarily close to the original N-dimensional projection. Therefore we also
have that
(29) E
HF
(N,Z)=inf

E
HF
(γ):γ
∗
= γ, γ
2
= γ, Trγ ≤ N

.
This Hartree-Fock minimization problem was studied by Lieb and Simon
in [16]. They proved the following about the existence of minimizers.
Theorem 3.4 (Existence of HF minimizers). If N is a positive integer
such that N<Z+1then there exists an N-dimensional projection
γ
HF
mini-
mizing the functional E
HF
in (26), i.e., E
HF
(N,Z)=E
HF
(
γ
HF
) is a minimum.
In the opposite direction the following result was proved by Lieb [13].
Theorem 3.5 (Lieb’s bound on the maximal ionization). If N is a
positive integer such that N>2Z +1 there are no minimizers for the Hartree-
Fock functional among N-dimensional projections, i.e., there does not exist an
N-dimensional projection γ such that E
HF
(γ)=E
HF
(N,Z).
This theorem will, in fact, follow from the proof of Lemma 7.1 below (see
page 503). Although this result is very good for Z =1it is far from optimal
for large Z.Inparticular the factor 2 should rather be 1. This fact known as
the ionization conjecture is one of the of the main results of the present work.
520 JAN PHILIP SOLOVEJ
Theorem 3.6 (Universal bound on the maximal ionization charge). There
exists a universal constant Q>0 such that for all positive integers satisfying
N ≥ Z + Q there are no minimizers for the Hartree-Fock functional among
N-dimensional projections.
Remark 3.7. Although, it is possible to calculate an exact value for the
constant Q above it is quite tedious to do so. Moreover, the present work
does not attempt to optimize this constant. The result of this work is mainly
to establish that such a finite constant exists. This of course raises the very
interesting question of finding a good estimate on the constant, but we shall
not address this here.
The proof of Theorem 3.6 is given in Section 13 on page 534.
Theorem 3.8 (Bound on the ionization energy). The ionization energy
of a neutral atom E
HF
(Z −1,Z)−E
HF
(Z, Z) is bounded by a universal constant
(in particular, independent of Z).
This theorem is proved in Section 13 on page 573.
The variational equations (Euler-Lagrange equations) for the minimizer
was also given in [16]. Since the Hartree-Fock variational equations shall be
used later in this work, we shall derive them in Theorem 3.11 below.
We first note that the Hartree-Fock functional E
HF
may be extended from
projections (i.e., density matrices with γ
2
= γ)toall density matrices. If
Tr [−∆γ] < ∞ all the terms of E
HF
are finite. In fact, Tr

Z|x|
−1
γ

is finite
by the Lieb-Thirring inequality (21) since Z|x|
−1
∈ L
∞
(
3
)+L
5/2
(
3
). The
term D(γ)isfinite by the Hardy-Littlewood-Sobolev inequality since
ρ
γ
∈
L
1
(
3
) ∩ L
5/3
(
3
) ⊂ L
6/5
(
3
). Finally, EX(γ) ≤ D(γ) since
D(γ) −E
X (γ)=
1
4

i,j
ν
i
ν
j

u
i
(x) ⊗ u
j
(y) −u
j
(x) ⊗ u
i
(y)
2
2
⊗
2
|x − y|
dx dy ≥ 0,
when ν
i
are the eigenvalues of γ with u
i
being the corresponding eigenfunctions.
If Tr [−∆γ]=∞ we set E
HF
(γ):=∞.Itisclear that lim
n
E
HF
(γ
n
)=∞ if
lim
n
Tr [−∆γ
n
] →∞.
Remark 3.9. It is important to realize that although D(γ) −E
X (γ)is
positive it is not a convex functional on the set of density matrices. In partic-
ular, the Hartree-Fock minimizer need not be unique. (A simple example of
nonuniqueness occurs for the case N =1.For a one-dimensional projection γ,
it is clear that D(γ) −E
X (γ)=0,hence the minimizer in this case is simply
the projection onto a ground state of the operator −
1
2
∆ −Z|x|
−1
on the space
L
2
(
3
;
2
). There are many ground states since the spin can point in any
direction.)
THE IONIZATION CONJECTURE IN HARTREE-FOCK THEORY 521
Another fact related to the nonconvexity of the Hartree-Fock functional
is the important observation first made by Lieb in [11] that the infimum of
the Hartree-Fock functional is not lowered by extending the functional to all
density matrices. For a simple proof of this see [1].
Theorem 3.10 (Lieb’s variational principle). Forall nonnegative inte-
gers N we have
inf

E
HF
(γ):γ
∗
= γ, γ = γ
2
, Tr[γ]=N

= inf {E
HF
(γ):0≤ γ ≤ I, Tr[γ]=N }
and if the infimum over all density matrices (the inf on the right) is attained
then so is the infimum over projections (the inf on the left).
We now come to the properties of the Hartree-Fock minimizers, especially
that they satisfy the Hartree-Fock equations. These equations state that a min-
imizing N-dimensional projection
γ
HF
is the projection onto the N-dimensional
space spanned by eigenfunctions with lowest possible eigenvalues for the HF
mean field operator
(30) H
γ
HF
:= −
1
2
∆ − Z|x|
−1
+
ρ
HF
∗|x|
−1
−K
γ
HF
.
Here K
γ
HF
is the exchange operator defined by having the 2 ×2-matrix valued
integral kernel
K
γ
HF
(x, y):=|x − y|
−1
γ
HF
(x, y).
Thus
γ
HF
(x, y)=

N
i=1
u
i
(x)u
i
(y)
∗
, where H
γ
HF
u
i
= ε
i
u
i
, and ε
1
,ε
2
, ,ε
N
≤ 0 are the N lowest eigenvalues of H
γ
HF
counted with multiplicities.
This self-consistent property of a minimizer
γ
HF
may equivalently be stated
as in the theorem below.
Theorem 3.11 (Properties of HF minimizers). If
γ
HF
with density
ρ
HF
is a
projection minimizing the HF functional E
HF
under the constraint Tr [
γ
HF
]=N
then
ρ
HF
∈ L
5/3
(
3
) ∩ L
1
(
3
) and H
γ
HF
defines a semibounded self -adjoint
operator with form domain H
1
(
3
;
2
) having at least N nonpositive eigen-
values. Moreover,
γ
HF
is the N-dimensional projection minimizing the map
γ → Tr

H
γ
HF
γ

.
Remark 3.12. The reader may worry that, because of degenerate eigen-
values of H
γ
HF
, the N -dimensional projection γ minimizing Tr

H
γ
HF
γ

may
not be unique. That it is, indeed, unique was proved in [2].
Proof of Theorem 3.11. We note that Tr [
γ
HF
]=N,Tr[−∆
γ
HF
] < ∞,
and the Lieb-Thirring inequality (21) implies that
ρ
HF
∈ L
5/3
(
3
) ∩ L
1
(
3
).
From this it is easy to see that
ρ
HF
∗|x|
−1
is a bounded function (in fact, it