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36 M. LEWIN, P. T. NAM, S. SERFATY, AND J. P. SOLOVEJ
In the latter bound, by choosing M = N 1/3 , we get
λL( � HN) ≥ λL(H)−C(εR(N)+N −1/3 ).
Spectral gap. By (i) one has for every L = 1,2,...,
liminf
N→∞ (λL(HN)−λ1(HN)) = λL(H)−λ1(H)
Taking the limit as L → ∞ we obtain the spectral gap
liminf
N→∞ (infσess(HN)−λ1(HN)) ≥ infσess(H)−λ1(H).
Step 2. Convergence of lower eigenvectors. First we consider the conver-
gence of ground states. If ΨN is a ground state of HN, then ΦN := UNΨN
on F≤N + . From the proof in
is a ground state of � HN := UN(HN −NeH)U ∗ N
Step 1, if max{NεR(N), √ N} ≪ M ≪ N, then we have ||gMΦN|| 2 → 0 and
lim
N→∞ 〈fMΦN,HfMΦN〉
�
= λ1(H) = Φ (1) ,HΦ (1)�
where Φ (1) is the unique ground state of H on F+.
To prove ΦN → Φ (1) , it suffices to show that fMΦN → Φ (1) . Let us write
fMΦN = aN +bN
where aN ∈ Span({Φ (1) }) and bN⊥Φ (1) . Then
〈fMΦN,HfMΦN〉 = 〈aN,HaN〉+〈bN,HbN〉 ≥ λ1(H)||aN|| 2 +λ2(H)||bN|| 2
= ||fMΦN|| 2 λ1(H)+(λ2(H)−λ1(H))||bN|| 2 .
Since 〈fMΦN,HfMΦN〉� ΦN − ||fMΦN|| 2 λ1(H) → 0 and λ2(H) > λ1(H), we
conclude that bN → 0 as N → ∞. Therefore, fMΦN → Φ (1) , and hence
ΦN → Φ (1) , as N → ∞.
Remark on the convergence in quadratic domain. We can show that if (A3s)
holds true, then we have the strong convergence ΦN → Φ (1) in the norm inducedbythequadraticformofHonF+,
namely〈ΦN,HΦN〉 → � Φ (1) ,HΦ (1)� .
In fact, from the above proof we already had
〈fMΦN,HfMΦN〉 →
�
Φ (1) ,HΦ (1)�
and
�
gMΦN, � �
HNgMΦN → 0.
On the other hand, from (A3s) and (16), we get � HN ≥ c0(H+N+)+g(N)
on F ≤N
+ , where c0 > 0 and g(N) → 0 as N → ∞. By choosing M such that
max{g(N),NεR(N), √ N} ≪ M ≪ N, we obtain
gM � HNgM ≥ c0(H+M)+g(N) ≥ c0H.
It implies that 〈gMΦN,HgMΦN〉 → 0. By using the localization for H in
Proposition 23 we can conclude that 〈ΦN,HΦN〉 → � Φ (1) ,HΦ (1)� .
Theconvergence of excited states of HN can beproved by usingthe above
argument and the following abstract result.
Lemma 25 (Convergence of approximate eigenvectors). Assume that A is
a self-adjoint operator, which is bounded from below, on a (separable) Hilbert
space, with the min-max values λ1(A) ≤ ... ≤ λL(A) < infσess(A). If the
normalized vectors {x (j)
n } L≥j≥1
n≥1 satisfy, for all i,j ∈ {1,2,...,L},
lim
n→∞ 〈x(i) n ,x(j) n 〉 = δij and lim
n→∞ 〈x(j) n ,Ax(j) n
〉 = λj(A),