Today Zhenghe explained Anderson Localizaion to me.

(update according to Zhenghe’s comment)

Consider a Schrodinger operator . Let and be the subspace spanned by all eigenvectors of .

The Schrodinger operator is said to displays *Anderson Localization* if

1. the eigenvectors span the whole space: (evidently this is stronger than ).

2. for each with the nontrivial eigenvector there exist and such that for each .

Note that for cocycle, if an e.v. has some which decays exponentially, then this e.v. must be simple.

It is quite interesting. The exponential decayed eigenvector corresponds to an orbit of a Schrodinger cocycle. Hence there exists a vector that is exponentially contracted under both forward and backward iterates. This resembles the case that a vector lies in the homoclinic tangency of a hyperbolic fixed point. This is an obstacle of uniform hyperbolicity. So positive Lyapunov exponent could only coexists with nonunifom hyperbolicity.