We treat first the case in whichhx=±1 for all sitesx and we introduce a unitary dressing transformation to control the spectrum for λ small. Then, we consider a situation in which |hx| can be less than one forx in a finite setL and prove exponential decay away fromL of dressed eigenfunctions with energy below the one-quasiparticle threshold. If the ground state is separated by a finite gap from the rest of the spectrum, this result can be strengthened and one can compute a second unitary transformation that makes the ground state of compact support. Finally, a case in which the singular setL is of finite density, is considered. The main technical tools we use are decay estimates on dressed Green's functions and variational inequalities.