Random Schrödinger operators with imaginary vector potentials are studied in dimension one. These operators are non-Hermitian and their spectra lie in the complex plane. We consider the eigenvalue problem on finite intervals of length $n$ with periodic boundary conditions and describe the limit eigenvalue distribution when $n$ goes to infinity. We prove that this limit distribution is supported by curves in the complex plane. We also obtain equations for these curves and for the corresponding eigenvalue density in terms of the Lyapunov exponent and the integrated density of states of a "reference" symmetric eigenvalue problem. In contrast to these results, the spectrum of the limit operator in $\ell^2(Z)$ is a two dimensional set which is not approximated by the spectra of the finite-interval operators.
Ilya Goldsheid. Boris Khoruzhenko. "Eigenvalue Curves of Asymmetric Tridiagonal Matrices." Electron. J. Probab. 5 1 - 28, 2000. https://doi.org/10.1214/EJP.v5-72