Distribution of eigenvalues in non-Hermitian Anderson models

We develop a theory which describes the behavior of eigenvalues of a class of one-dimensional random non-Hermitian operators introduced recently by Hatano and Nelson. We prove that the eigenvalues are distributed along a curve in the complex plane. An equation for the curve is derived, and the density of complex eigenvalues is found in terms of spectral characteristics of a "reference" Hermitian disordered system. The generic properties of the eigenvalue distribution are discussed.