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

REMARK 5.21. Above we derived (5.53) only for z ∉ σ(H Λ2 \Λ 1
). However, both
sides of (5.53) exist and are analytic outside σ(H Λ1 ) ∪ σ(H Λ2 ), so the formula is
valid for all z outside σ(H Λ1 ) ∪ σ(H Λ2 )
We introduce a short-hand notation for the matrix elements of resolvents
G Λ z (n, m) = (H Λ − z) −1 (n, m). (5.54)
The functions G Λ z are called Green’s functions .
With this notation the geometric resolvent equation reads
∑
G Λ 2
z (n, m) = G Λ 1
z (n, k)G Λ 2
z (k ′ , m) . (5.55)
(k,k ′ )∈∂Λ 1
k∈Λ 1 , k ′ ∈Λ 2
There are analogous equations to (5.53) for Dirichlet or Neumann boundary conditions
which can be derived from (5.46) and (5.46) in the same way as above.
5.4. An alternative approach to the density of states.
In this section we present an alternative definition of the density of states measure.
Perhaps, this is the more traditional one. We prove its equivalence to the definition
given above.
In section 5.1 we defined the density of states measure by starting with a function
ϕ of the Hamiltonian, taking its trace restricted to a cube Λ L and normalizing
this trace. In the second approach we first restrict the Hamiltonian H ω to Λ L with
appropriate boundary conditions, apply the function ϕ to the restricted Hamiltonian
and then take the normalized trace.
For any Λ let H X Λ be either H Λ or H N Λ or HD Λ . We define the measures ˜νX L ( i.e.
˜ν L , ˜ν N L , ˜νD L ) by
∫
ϕ(λ) d˜ν X L (λ) = 1
|Λ L | tr ϕ(HX Λ L
) . (5.56)
Note that the operators H X Λ L
act on the finite dimensional Hilbert space l 2 (Λ L ), so
their spectra consist of eigenvalues E n (H X Λ L
) which we enumerate in increasing
order
E 0 (H X Λ L
) ≤ E 1 (H X Λ L
) ≤ . . . .
In this enumeration we repeat each eigenvalue according to its multiplicity (see
also (3.43).
With this notation (5.56) reads
∫
ϕ(λ) d˜ν L X (λ) = 1 ∑
ϕ(E n (HΛ X |Λ L |
L
)) .
The measure ˜ν
L X is concentrated on the eigenvalues of HX Λ L
. If E is an eigenvalue
of HΛ X L
then ˜ν
L X ({E}) is equal to the dimension of the eigenspace corresponding
to E.
n
41