"Lifshitz tails" and extended states in an Imaginary random potential
  Non-Hermitean operators may appear during the calculation of a partition function in various models of statistical mechanics. The tail eigen-states, having anomalously small real part of energy $Re(\eps)$, became naturally important in this case. We consider the distribution of such states and the form of eigenfunctions for the particle propagating in an imaginary random potential (the model motivated by the statistics of polymer chains). Unlike it is in the Hermitean quantum mechanics, our tail states are sufficiently extended. Such state appear if the values of random potential turns out to be anomalously close inside the large area. Results of numerical simulations in the case of strong coupling confirm the analytic estimates. 
P. G. Silvestrov