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

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(H 0 u)(n) = − ∑
|| m−n|| 1 =1
(u(m) − u(n)) . (3.7)
This operator is also known as the graph Laplacian for the graph Z d or the discrete
Laplacian. Its quadratic form is given by
〈u, H 0 v〉 = 1 ∑ ∑
(u (n) − u (m)) (v(n) − v(m)) . (3.8)
2
n∈Z d
|| m−n|| 1 =1
We call this sesquilinear form a Dirichlet form because of its similarity to the classical
Dirichlet form
∫
〈u, −△ v〉 = ∇u(x) · ∇v(x) dx .
R d
The operator H 0 is easily seen to be symmetric and bounded, in fact
( ∑ ( ∑
‖H 0 u‖ =
n∈Z d
|| j || 1 =1
(
u(n + j) − u(n)
) ) 2) 1
2
(3.9)
≤
∑ ( ∑
| u(n + j) − u(n) | 2) 1 2
(3.10)
|| j || 1 =1 n∈Z d
≤
∑ ( ∑ ( ) ) 1
2 2
| u(n + j) | + | u(n) | (3.11)
|| j|| 1 =1 n∈Z d
≤
∑ ( ( ∑
) 1 ( ∑
| u(n + j) | 2 2
+ | u(n) | 2) )
1
2
(3.12)
|| j|| 1 =1 n∈Z d n∈Z d
≤ 4 d ‖ u ‖ . (3.13)
From line (3.9) to (3.10) we applied the triangle inequality for l 2 to the functions
∑
fj (n) with f j (n) = u(n + j) − u(n). In (3.12) and (3.13) we used the triangle
inequality and the fact that any lattice point in Z d has 2d neighbors.
Let us define the Fourier transform from l 2 (Z d ) to L 2 ([0, 2π] d ) by
(Fu)(k) = û(k) = ∑ n
u n e −i n·k . (3.14)
F is a unitary operator. Under F the discrete Laplacian H 0 transforms to the
multiplication operator with the function h 0 (k) = 2 ∑ d
ν=1 (1 − cos(k ν)), i.e.
FH 0 F −1 is the multiplication operator on L 2 ([0, 2π] d ) with the function h 0 . This
shows that the spectrum σ(H 0 ) equals [0, 4d] (the range of the function h 0 ) and
that H 0 has purely absolutely continuous spectrum.
It is very convenient that the ‘discrete Dirac function’ δ i defined by (δ i ) j = 0 for
i ≠ j and (δ i ) i = 1 is an ‘honest’ l 2 -vector, in fact the collection {δ i } i∈Z d is an