An Invitation to Random Schr¨odinger operators - FernUniversität in ...

Analogously we get
∫
N(E) − χ (−∞,E] (λ) dν L (λ) (5.17)
≥ N(E − ) − N(E + ) − ∣ ∫
∣N(E + ) − χ (−∞,E+ ](λ) dν L (λ) ∣ (5.18)
≥ −ε . (5.19)
33
Hence
∫
∣ N(E) −
χ (−∞,E] (λ) dν L (λ) ∣ ∣ → 0
This proves (5.12) for continuity points. Since there are at most countably many
points of discontinuity for N another application of Proposition 5.2 proves the
result for all E.
□
Above we used the following Lemma.
LEMMA 5.10. If the function F : R → R is monotone increasing then F has at
most countably many points of discontinuity.
PROOF: Since F is monotone both F (t−) = lim s↗t F (s) and F (t+) = lim s↘t F (s)
exist. If F is discontinuous at t ∈ R then F (t+) − F (t−) > 0. Set
D n = {t ∈ R | F (t+) − F (t−) > 1 n }
then the set D of discontinuity points of F is given by ⋃ n∈N D n.
Let us assume that D is uncountable. Then also one of the D n must be uncountable.
Since F is monotone and defined on all of R it must be bounded on any bounded
interval. Thus we conclude that D n ∩ [−M, M] is finite for any M. It follows
that D n = ⋃ (
M∈N Dn ∩ [−M, M] ) is countable. This is a contradiction to the
conclusion above.
□
REMARK 5.11.
The proof of Corollary 5.8 shows that we also have
N(E−) = sup N(E − ε)
=
ε>0
∫
χ (−∞,E) (λ) dν(λ)
∫
= lim
L→∞
χ (−∞,E) (λ) dν L (λ) (5.20)
for all E and P-almost all ω (with an E-independent set of ω).
Consequently, we also have ν({E}) = lim L→∞ ν L ({E}).
PROPOSITION 5.12. supp(ν) = Σ (= σ(H ω )).