Resonances in a one-dimensional disordered chain

We study the average density of resonances, ⟨ρ( x , y )⟩, in a semi-infinite disordered chain coupled to a perfect lead. The function ⟨ρ( x , y )⟩ is defined in the complex energy plane and the distance y from the real axes determines the resonance width. We concentrate on strong disorder and derive the asymptotic behaviour of ⟨ρ( x , y )⟩ in the limit of small y .