Calculation of Anderson localization criterium for a one dimensional chain with diagonal disorder

For a one dimensional half-infinite chain with diagonal disorder we calculated the ultimate at $t\to\infty$ value of the average excitation density at the edge site $D$ if at $t=0$ the excitation was localised at the edge site (Anderson' s creterium). We obtained the following results: i) for the binary disordered chain we derived the close expression for $D$ which is exact in the limit of low concentration of defects and is valid for an arbitrary energy of defects $\varepsilon$. In this case $D$ demonstrated the non analytical dependence on $\varepsilon$. ii) The close expression for $D$ is obtained for the case of an arbitrary small disorder. iii) The relative contribution of states with specified energy to $D$ is calculated. All the results obtained are in complete agreement with computer simulation.