The semi-circle diagrams in the stochastic limit of the Anderson model

We prove that, in the stochastic limit of the Anderson model only the non-c rossing diagrams survive for the transition amplitude from the first excite d state of the free Hamiltonian to the first excited state of the interacti ng Hamiltonian. This confirms a conjecture of Migdal (1958) and Abrikosov, Gorkov, Dzyaloshinski (1975). From this we deduce a closed (nonlinear) Schw inger-Dyson type equation for the limit transition amplitude whose solution can be found and gives the explicit dependence of this amplitude on the mo mentum of the excited state.