Localization in an imaginary vector potential

Eigenfunctions of 1d disordered Hamiltonian with constant imaginary vector potential are investigated. Even within the domain of complex eigenvalues the wave functions are shown to be strongly localized. However, this localization is of a very unusual kind. The logarithm of the wave function at different coordinates $x$ fluctuates strongly (just like the position of Brownian particle fluctuates in time). After approaching its maximal value the logarithm decreases like the square root of the distance $\bar{(\ln|\psi_{max}/\psi|)^2} \sim |x-x_0|$. The extension of the model to the quasi-1d case is also considered.