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

45
is differentiable as well. Furthermore, since
H Λ (V Λ ) − E = H Λ (V Λ − E) . (5.66)
it follows
tr ϱ ( H Λ (V Λ ) − E ) = F ( {V i − E} i∈Λ
)
(5.67)
and consequently
∂
∂E
(
tr ϱ ( H Λ (V Λ ) − E )) = − ∑ i∈Λ
∂
(
tr ϱ ( H Λ (V Λ ) − E )) (5.68)
∂V i
Therefore, with (5.64)
N Λ (E + ε) − N Λ (E − ε) ≤ tr ϱ (H Λ − E + 2ε) − tr ϱ (H Λ − E − 2ε)
= − ( tr ϱ (H Λ − (E + 2ε)) − tr ϱ( H Λ − (E − 2ε)) )
= −
=
∫ E+2ε
E−2ε
∫ E+2ε
E−2ε
∂
∂η
∑
j∈Λ
(
ϱ ( H Λ (V Λ ) − η )) dη
∂
∂V j
tr ϱ ( H Λ (V Λ − η) ) dη. (5.69)
Therefore
E ( N Λ (E + ε) − N Λ (E − ε) )
≤ E ( ∑
= ∑ j∈Λ
∫ E+2ε
j∈Λ
E−2ε
∫ E+2ε
E−2ε
∂
∂V j
tr (ϱ (H Λ (V Λ ) − η) ) dη)
( ∂
E tr ( ϱ (H Λ (V Λ ) − η) )) dη. (5.70)
∂V j
Since the random variables V ω (i) are independent and have the common distribution
dP 0 (V i ) = g(V i ) dV i , the expectation E is just integration with respect to
the product of these distributions. Moreover, since supp P 0 is compact the integral
over the variable V i can be restricted to [−M, +M] for some M large enough.