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Lemma 8.2]
BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 21
2
π|x| ≤ √ −∆ ≤ √ 1−∆ on L 2 (R 3 ),
while if K = −∆, then we even have the stronger bound
|w(x−y)| 2 ≤ C0[Kx +Ky +V(x)+V(y)]+C on L 2 ((R d ) 2 ) (33)
due to Hardy’s inequality (29). When d = 2, then (33) also holds true, due
to the estimates K+V ≥ C−1 [log(1+|x|)] 2 −C and
|w(x−y)| 2 �
1
≤ 2
|x−y| +[log(1+|x|)]2 +[log(1+|y|)] 2
�
+C
for some C > 0. Thus (A1) holds true. On the other hand, Assumption
(A2) follows from the following
Proposition 7 (Hartree theory). Under the previous assumptions, the variational
problem
eH := inf
u∈L2 (Rd �
〈u,(K+V)u〉+
)
||u||=1
1
� �
|u(x)|
2
2 w(x−y)|u(y)| 2 �
dxdy (34)
has a unique minimizer u0 which satisfies that u0(x) > 0 for a.e. x ∈ Rd and solves the mean-field equation
⎧
⎨hu0
= 0,
⎩h
:= K+V +|u0| 2 ∗w−µH,
for some Lagrange multiplier µH ∈ R. The operator h has only discrete
spectrum λ1(h) < λ2(h) ≤ λ3(h) ≤ ... with limi→∞λi(h) = ∞. Moreover,
the operator K with kernel K(x,y) = u0(x)w(x−y)u0(y) is Hilbert-Schmidt
on L 2 (R d ) and it is positive on H+. Finally, u0 is non-degenerate in the
sense of (7).
Before proving Proposition 7, let us mention that in 2D, the Coulomb
potential w(x) = −log|x| does not have positive Fourier transform. More
precisely, �w = pv | · | −1 , the principal value of | · | −1 . Although w is not
a positive type kernel, we still have the following restricted positivity (see
[13]).
Proposition 8 (Coulomb log kernel). For any function f ∈ L1 (R2 ) ∩
L1+ε (R2 ) for some ε > 0 with
�
�
log(2+|x|)|f(x)|dx < ∞ and f = 0,
we have
R 2
��
0 ≤ D(f,f) := −
We can now provide the
R 2 ×R 2
R 2
f(x)log|x−y|f(y)dxdy < ∞.