Weak disorder expansions for the Anderson model on a one dimensional strip at the center of the band
Let \(H_ \lambda=-\Delta/2+\lambda V\) be a family of random Schrödinger operators on a discrete strip of width \(l\), indexed by the real parameter \(\lambda\), such that the random potentials \(V\) are i.i.d. at each site \(x=(n,i)\); \(n\in\mathbb{Z}\); \(i=1,\dots,l\). For a real energy \(E\) one considers the boundary value of the averaged Green matrix defined by \[ J(\lambda,E)=\lim_{\varepsilon\to 0}\mathbb{E}\{G_ \lambda(E+i\varepsilon,n,n)\}. \] It can be proved that under some mild conditions on the probability distribution of the potential this limit exists for all \((E,\lambda)\) and is a smooth function of \(E\) for \(\lambda\neq 0\). The problem is to find an asymptotic expansion of \(J(\lambda,E)\) when \(\lambda\to 0\). In one dimension (that is for \(l=1)\) it has already been proved that such an expansion is in general valid but is inadequate for some special values of the energy \(E\). The same result is proved here, by using the supersymmetric method in order to rewrite the Green matrix in a suitable way, but the definition of the exceptional energies have to be adapted to this situation. This result can be used to obtain similar expansions for the density of states and the sum of the nonnegative Lyapunov exponents.
probability distribution of the potential
scientific article; zbMATH DE number 97872