Regular Spacings of Complex Eigenvalues in the One-Dimensional Non-Hermitian Anderson Model

Abstract

We prove that in dimension one the non-real eigenvalues of the non-Hermitian Anderson (NHA) model with a selfaveraging potential are regularly spaced. The class of selfaveraging potentials which we introduce in this paper is very wide and in particular includes stationary potentials (with probability one) as well as all quasi-periodic potentials. It should be emphasized that our approach here is much simpler than the one we used before. It allows us a) to investigate the above mentioned spacings, b) to establish certain properties of the integrated density of states of the Hermitian Anderson models with selfaveraging potentials, and c) to obtain (as a by-product) much simpler proofs of our previous results concerned with non-real eigenvalues of the NHA model.

Real eigenvalues in the non-Hermitian Anderson model

“…The numerical work of Hatano and Nelson suggests that all of the eigenvalues remain in the real axis for sufficiently small 0 ≤ g < g 1 ; all eigenvalues move out of the real axis and align along a smooth curve on the complex plane for sufficiently large g > g 2 ; and we see a combination of the two for g 1 < g < g 2 . For a more general setting which includes (0.1) as a special case, Goldsheid and Khoruzhenko [GoKh98,GoKh00,GoKh03] showed that the behaviour of the eigenvalues of (0.1) depends on the Lyapunov exponent associated with the Hermitian operator,…”