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

If H = H 0 + V we define H Λ = (H 0 ) Λ + V , where in the latter expression
V stands for the multiplication with the function V restricted to Λ. Similarly,
HΛ N = (H 0) N Λ + V and HD Λ = (H 0) D Λ + V .
These operators satisfy
H N Λ ⊕ H N ∁Λ ≤ H ≤ HD Λ ⊕ H D ∁Λ . (5.42)
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For Λ 1 ⊂ Λ ⊂ Z d we have analogs of the ‘splitting’ formulae (5.31), (5.37) and
(5.38). To formulate them it will be useful to define the ‘relative’ boundary ∂ Λ2 Λ 1
of Λ 1 ⊂ Λ 2 in Λ 2 .
∂ Λ2 Λ 1 = ∂Λ 1 ∩ (Λ 2 × Λ 2 ) = ∂Λ 1 \ ∂Λ 2 (5.43)
= { (i, j) ∣ ∣ || i − j|| 1 = 1 and i ∈ Λ 1 , j ∈ Λ 2 \ Λ 1 or i ∈ Λ 2 \ Λ 1 , j ∈ Λ 1
}
The analogs of the splitting formulae are
with
H Λ2 = H Λ1 ⊕ H Λ2 \Λ 1
+ Γ Λ 2
Λ 1
(5.44)
H N Λ 2
= H N Λ 1
⊕ H N Λ 2 \Λ 1
+ Γ Λ 2 N
Λ 1
(5.45)
H D Λ 2
= H D Λ 1
⊕ H D Λ 2 \Λ 1
+ Γ Λ 2 D
Λ 1
(5.46)
⎧
⎪⎨
Γ Λ 2
Λ 1
(i, j) =
⎪⎩
⎧
⎪⎨
Λ 1
(i, j) =
⎪⎩
Γ Λ 2 N
⎧
⎪⎨
Λ 1
(i, j) =
⎪⎩
Γ Λ 2 D
0 if i = j and i ∈ Λ 1
0 if i = j and i ∈ Λ 2 \ Λ 1
−1 if (i, j) ∈ ∂ Λ2 Λ 1
0 otherwise.
n Λ2 (i) − n Λ1 (i) if i = j and i ∈ Λ 1
n Λ2 (i) − n Λ2 \Λ 1
(i) if i = j and i ∈ Λ 2 \ Λ 1
−1 if (i, j) ∈ ∂ Λ2 Λ 1
0 otherwise.
n Λ1 (i) − n Λ2 (i) if i = j and i ∈ Λ 1
n Λ2 \Λ 1
(i) − n Λ2 (i) if i = j and i ∈ Λ 2 \ Λ 1
−1 if (i, j) ∈ ∂ Λ2 Λ 1
0 otherwise.
(5.47)
(5.48)
(5.49)
In particular, for Λ = Λ 1 ∪ Λ 2 with disjoint sets Λ 1 and Λ 2 we have
H N Λ 1
⊕ H N Λ 2
≤ H N Λ ≤ H D Λ ≤ H D Λ 1
⊕ H D Λ 2
. (5.50)
since Γ Λ N
Λ 1
≥ 0 and Γ Λ D
Λ 1
≤ 0.