Statistics of resonances in a semi-infinite disordered chain

We study the average density of resonances (DOR) for a semi-infinite disordered chain, coupled to the outside world by a (semi-infinite) perfect lead. A set of equations is derived, which provides the general framework for calculating the average DOR, for an arbitrary disorder and coupling strength. These general equations are applied to the case of weak coupling and an asymptotically exact expression for the averaged DOR is derived, in the limit of small resonance width. This expression is universal, in the sense that it holds for any degree of disorder and everywhere in the (unperturbed) energy band.