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

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5.1. Definition and existence.
5. The density of states
Here, as in the rest of the paper we consider the Anderson model, i.e.
H ω = H 0 + V ω on l 2 (Z d ) with independent random variables V ω (n) with a common
distribution P 0 .
In this section we define a quantity of fundamental importance for models in
condensed matter physics: the density of states. The density of states measure
ν([E 1 , E 2 ]) gives the ‘number of states per unit volume’ with energy between E 1
and E 2 . Since the spectrum of our Hamiltonian H ω is not discrete we can not
simply count eigenvalues within the interval [E 1 , E 2 ] or, what is the same, take the
dimension of the corresponding spectral projection. In fact, the dimension of any
spectral projection of H ω is either zero or infinite. Instead we restrict the spectral
projection to the finite cube Λ L (see 3.5) in Z d , take the dimension of its range
and divide by | Λ L | = (2L + 1) d the number of points in Λ L . Finally, we send
the parameter L to infinity. This procedure is sometimes called the thermodynamic
limit.
For any bounded measurable function ϕ on the real line we define the quantity
ν L (ϕ) = 1
|Λ L | tr ( χ Λ L
ϕ(H ω ) χ ΛL ) = 1
|Λ L | tr ( ϕ(H ω) χ ΛL ) . (5.1)
Here χ Λ denotes the characteristic function of the set Λ, (i.e. χ Λ (x) = 1 for
x ∈ Λ and = 0 otherwise). The operators ϕ(H ω ) are defined via the spectral
theorem (see Section 3.2). In equation (5.1) we used the cyclicity of the trace, (i.e.:
tr(AB) = tr(BA)) and the fact that χ 2 Λ = χ Λ .
Since ν L is a positive linear functional on the bounded continuous functions, by
Riesz representation theorem, it comes from a measure which we also call ν L , i.e.
∫
ν L (ϕ) = ϕ(λ) dν L (λ). (5.2)
R
We will show in the following that the measures ν L converge to a limit measure ν
as L → ∞ in the sense of vague convergence of measures for P-almost all ω.
DEFINITION 5.1. A series ν n of Borel measures on R is said to converge vaguely
to a Borel measure ν if ∫
∫
ϕ(x) d ν n (x) →
ϕ(x) d ν(x)
for all function ϕ ∈ C 0 (R), the set of continuous functions with compact support.
We start with a proposition which establishes the almost sure convergence of the
integral of ν L over a given function.
PROPOSITION 5.2. If ϕ is a bounded measurable function, then for P-almost all ω
lim
L→∞
1
|Λ L | tr ( ϕ(H ω ) χ ΛL
)
= E
(
〈δ0 , ϕ(H ω )δ 0 〉 ) . (5.3)