  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.
