For the wave equation, the propagator in the Fourier domain is written as G(k,ω)=−1ω2c2−k2+iϵ.
When
ω/c is close to
∥k∥=k, we have the following approximation (Stokholski-Plemelj theorem)
1πImG(k,ω)=−δ(ω2c2−k2)=−c22ωδ(ω−ck),
such that, if we integrate over
k we get the famous expression of the density of states
ρ(ω)=−2ωπc2ImG(r=0,ω).
Can one do the same computation with the diffusive propagator (diffuson) G(k,ω)=1iωD−k2?
If ω is close to zero we have 1πImG(k,ω=0)=δ(k2)
or taking the real part
1πReG(k=0,ω)=Dδ(ω),
but I can't really see how one could obtain a relation such as (1).
EDIT
Considering that ImG(r=0,ω)=−14π√ω2D
in three dimensions, it is tempting to use a similar relation to get
ρdiff3(ω)=14π2√ω2D3.
But is this really justified ?
This post imported from StackExchange Physics at 2015-06-17 16:39 (UTC), posted by SE-user Tom-Tom