Computation of localization degree in the sense of the Anderson criterion for a one-dimensional diagonally disordered system

For a one-dimensional diagonally disordered half-infinite chain, we consider the problem of finding the limit value as $t\to\infty$ of the average excitation density $D$ at the edge site of the chain under the condition that the excitation is localized at this site at $t=0$. For a binary disordered chain, we obtain an expression for $D$ that is exact in the small defect concentration limit for an arbitrary defect energy. In this case, the density $D$ depends nonanalytically on the energy. We obtain an expression for $D$ in the case of an arbitrary small diagonal disorder. We also calculate the relative contribution to $D$ from states with a given energy. All the obtained results agree well with the computer simulation data.