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28 M. LEWIN, P. T. NAM, S. SERFATY, AND J. P. SOLOVEJ
Proof of Corollary 16. It suffices to show the statement for Φ ′ = Φ. Note
that the quadraticform domain of H is thesame as that of dΓ(h+1) because
of (16), and dΓ(h + 1) preserves all the subspaces Hm +’s. Therefore, if we
denote by ΦM the projection of Φ onto F ≤M
+ , then
lim
M→∞ 〈H〉ΦM = 〈H〉Φ and lim
M→∞ lim 〈H+C〉ΦN−ΦM = 0.
N→∞
If we denote � HN := UN(HN − N eH)U∗ N , then from Proposition 15 we
have
lim
M→∞ 〈� HN −H〉ΦM
= 0 and lim
M→∞ lim
N→∞ 〈� HN〉ΦN−ΦM
= 0.
The latter convergence still holds true with � HN replaced by a non-negative
operator H ′ N := � HN+C0(H+C0), where C0 > 0 is chosen large enough. By
using the Cauchy-Schwarz inequality for the operator H ′ N , we deduce that
lim
M→∞ lim
N→∞ 〈� HN〉Φ −〈 � HN〉ΦM
= 0.
We then can conclude that 〈 � HN〉Φ → 〈H〉Φ as N → ∞. �
The rest of the section is devoted to the proof of Proposition 15. We shall
need the following technical result.
Lemma 17. If (A1)-(A2) hold true, then we have the operator inequalities
on F+:
dΓ(QTQ) ≤
dΓ(Q(|u0| 2 ∗|w|)Q) ≤
1
H+CN+ +C,
1−α1
α2
H+CN+ +C,
1−α1
where 1 > α1 > 0 and α2 > 0 are given in the relative bound (5) in Assumption
(A1).
Proof. Using (5) we get |u0| 2 ∗w ≥ −α1(T +C) and |u0| 2 ∗|w| ≤ α2(T +C).
Consequently, T ≤ (1 − α1) −1 h + C and |u0| 2 ∗ |w| ≤ α2(1 − α1) −1 h + C.
The desired estimates follows from the lower bound H ≥ dΓ(h)−CN+−C
(see Remark 10). �
Now we give the
Proof of Proposition 15. Let Φ be a normalized vector in the quadratic form
domain of H such that Φ ∈ F ≤M
+ for some 1 ≤ M ≤ N. Starting from
the identity (44), we shall compare 〈A2〉Φ with 〈H〉Φ, and show that the
remaining part is negligible.
Step 1. Main part of the Hamiltonian.
Lemma 18 (Bound on A2 −H). We have
� �
� �
�〈A2〉Φ −〈H〉Φ�
≤ M
� �
α2
〈H〉Φ +C〈N+ +1〉Φ .
N −1 1−α1