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

37
⎧
⎨ H Λ (n, m) if n, m ∈ Λ,
(H Λ ⊕ H ∁Λ )(n, m) = H
⎩ ∁Λ (n, m) if n, m ∉ Λ,
0 otherwise.
(5.32)
In other words H Λ ⊕ H ∁Λ is a block diagonal matrix and Γ Λ is the part of H which
connects these blocks.
It is easy to see, that Γ Λ is neither negative nor positive definite. Consequently, the
operator H Λ will not satisfy any inequality of the type (5.24).
To obtain analogs to Dirichlet and Neumann boundary conditions we should substitute
the operator Γ Λ in (5.31) by a negative definite resp. positive definite operator
and H Λ ⊕ H ∁Λ by an appropriate block diagonal matrix.
For the operator H 0 the diagonal term H 0 (i, i) = 2d gives the number of sites j
to which i is connected (namely the 2d neighbors in Z d ). This number is called
the coordination number of the graph Z d . In the matrix H Λ the edges to ∁Λ are
removed but the diagonal still contains the ‘old’ number of adjacent edges. Let us
set n Λ (i) = | {j ∈ Λ| || j − i|| 1 = 1}| to be the number of sites adjacent to i in Λ,
the coordination number for the graph Λ. n Λ (i) = 2d as long as i ∈ Λ\∂ − Λ but
n Λ (i) < 2d at the boundary. We also define the adjacency matrix on Λ by
{
−1 if i, j ∈ Λ, || i − j||1 = 1
A Λ (i, j) =
0 otherwise.
(5.33)
The operator (H 0 ) Λ on l 2 (Λ) is given by
where 2d denotes a multiple of the identity.
(H 0 ) Λ = 2d + A Λ (5.34)
DEFINITION 5.16. The Neumann Laplacian on Λ ⊂ Z d is the operator on l 2 (Λ)
defined by
(H 0 ) N Λ = n Λ + A Λ . (5.35)
Above n Λ stands for the multiplication operator with the function n Λ (i) on l 2 (Λ).
REMARK 5.17.
(1) In (H 0 ) Λ the off diagonal term ‘connecting’ Λ to Z d \Λ are removed.
However, through the diagonal term 2d the operator still ‘remembers’
there were 2d neighbors originally.
(2) The Neumann Laplacian H N 0 on Λ is also called the graph Laplacian. It
is the canonical and intrinsic Laplacian with respect to the graph structure
of Λ. It ‘forgets’ completely that the set Λ is imbedded in Z d .
(3) The quadratic form corresponding to (H 0 ) N Λ
is given by
〈u, (H 0 ) N Λ v〉 = 1 ∑
(u(n) − u(m))(v(n) − v(m)) .
2
n,m∈Λ
|| n−m|| 1 =1