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

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(3) We denote by σ ac (H), σ sc (H), σ pp (H) the absolutely continuous (resp.
singularly continuous, resp. pure point) spectrum of the operator H. For
a definition and basic poperties we refer to Sections 7.2and 7.3.
PROOF (Sketch) : If H ω is ergodic and f is a bounded (measurable) function
then f(H ω ) is ergodic as well, i.e.
f(H Ti w) = U i f(H ω )U ∗ i .
(see Lemma 4.5).
We have (λ, µ) ∩ σ(H ω ) ≠ ∅ if and only if χ (λ,µ) (H ω ) ≠ 0.
This is equivalent to Y λ,µ (ω) := tr χ (λ,µ) (H ω ) ≠ 0.
Since χ (λ,µ) (H ω ) is ergodic, Y λ,µ is an invariant random variable and consequently,
by Proposition 4.1 Y λ,µ = c λ,µ for all ω ∈ Ω λ,µ with P (Ω λ,µ ) = 1.
Set
Ω 0 =
⋂
Ω λ,µ .
λ,µ∈Q, λ≤µ
Since Ω 0 is a countable intersection of sets of full measure, it follows that P (Ω 0 ) =
1. Hence we can set
Σ = {E | c λ,µ ≠ 0 for all λ < E < µ, λ, µ ∈ Q} .
To prove the assertions on σ ac we need that the projection onto H ac , the absolutely
continuous subspace with respect to H ω is measurable, the rest is as above. The
same is true for σ sc and σ pp .
We omit the measurability proof and refer to [64] or [23].
□
Above we used the following results
LEMMA 4.5. Let A be a self adjoint operators and U a unitary operator, then for
any bounded measurable function f we have
f ( UAU ∗) = U f(A) U ∗ . (4.7)
PROOF: For resolvents, i.e. for f z (λ) = 1
λ−z
with z ∈ C \ R equation (4.7)
can be checked directly. Linear combinations of the f z are dense in C ∞ (R), the
continuous functions vanishing at infinity, by the Stone-Weierstraß theorem (see
Section 3.3). Thus (4.7) is true for f ∈ C ∞ (R).
If µ and ν are the projection valued measures for A and B = UAU ∗ respectively,
we have therefore for all f ∈ C ∞ (R)
∫
f(λ) d ν ϕ,ψ (λ) = 〈ϕ, f(B) ψ〉
= 〈ϕ, Uf(A) U ∗ ψ〉
= 〈U ∗ ϕ, f(A) U ∗ ψ〉
∫
= f(λ) d µ U ∗ ϕ,U ∗ ψ(λ) (4.8)