Localization in Any Dimension

Abstract

Let β be a nonzero real number and let H(ω) be the random Schrodinger operator on ℤd defined by $$ H(\omega )\psi (x)\;{\rm{ = }}\;{H_0}\psi (x)\;{\rm{ + }}\;\beta V(x,\omega )\psi (x) $$ where {V(x) ; x ∈ ℤd} is an i.i.d. family of potentials defined on some probability space (Ω,ℙ). Such a model proposed by Anderson in [8], is usually refered to as the “Anderson model”. It has been shown in the preceding chapters that the behavior at infinity of the solutions of the “eigenvalue equation” $$ {H_0}\psi (x)\;{\rm{ = }}\;(\lambda \;{\rm{ - }}\;\beta V(x,\omega ))\psi (x) $$ is crucial in the study of spectral properties of the operator H(ω). The behavior at infinity of the Green’s function G(⋋,x,y) which satisfies the above equation for any x ≠ y is also of primordial interest. The links between the exponential growth of the solutions of the eigenvalue equation and the exponential decay of the Green’s function have already been pointed out in the one dimensional case.