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

PROOF (Lemma) :
We define ˜Λ L = {i ∈ Λ L | (L − 1) ≤ || i|| ∞ ≤ L }
˜Λ L consists of the two outermost layers of Λ L .
The values u(n) of an eigenfunction u of H ω with H ω u = λu can be computed
from the eigenvalue equation for all n ∈ Λ L once we know its values on ˜Λ L . So,
the dimension of χ ΛL (V λ ) is at most the number of points in ˜Λ L .
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5.2. Boundary conditions.
Boundary conditions are used to define differential operators on sets M with a
boundary. A rigorous treatment of boundary conditions for differential operators is
most conveniently based on a quadratic form approach (see [115]) and is out of the
scope of this review. Roughly speaking boundary conditions restrict the domain
of a differential operator D by requiring that functions in the domain of D have
a certain behavior at the boundary of M. In particular, Dirichlet boundary conditions
force the functions f in the domain to vanish at ∂M. Neumann boundary
conditions require the normal derivative to vanish at the boundary. Let us denote
by −∆ D M and −∆N M
the Laplacian on M with Dirichlet and Neumann boundary
condition respectively.
The definition of boundary conditions for the discrete case are somewhat easier
then in the continuous case. However, they are presumably less familiar to the
reader and may look somewhat technical at a first glance. The reader might therefore
skip the details for the first reading and concentrate of ‘simple’ boundary conditions
defined below. Neumann and Dirichlet boundary conditions will be needed
for this text only in chapter 6 in the proof of Lifshitz tails.
For our purpose the most important feature of Neumann and Dirichlet boundary
conditions is the so called Dirichlet-Neumann bracketing . Suppose M 1 and M 2
are disjoint open sets in R d and M = (M 1 ∪ M 2 ) ◦ , ( ◦ denoting the interior) then
−∆ N M 1
⊕ −∆ N M 2
≤ −∆ N M ≤ −∆ D M ≤ −∆ D M 1
⊕ −∆ D M 2
. (5.24)
in the sense of quadratic forms. In particular the eigenvalues of the operators in
(5.24) are increasing from left to right.
We recall that a bounded operator A on a Hilbert space H is called positive (or
positive definite or A ≥ 0) if
〈ϕ, A ϕ〉 ≥ 0 for all ϕ ∈ H . (5.25)
For unbounded A the validity of equation 5.25 is required for the (form-)domain
of A only.
By A ≤ B for two operators A and B we mean B − A ≥ 0.
For the lattice case we introduce boundary conditions which we call Dirichlet and
Neumann conditions as well. Our choice is guided by the chain of inequalities(5.24).