Consider the vector field


By homogeneity, one has



We are looking for eigenvalues close to -1; so, in the region of interest


The eigenvalues move when the random variables move along the vector field.

This is independent of the sign of the single site potential.

Restriction : energy free zone for the background operator.

One can also deal with internal gaps of

Then, one has to assume that the random perturbation is small.