  We construct efficient approximations for the eigenfunctions of
non-self-adjoint Schroedinger operators in one dimension. The same ideas also
apply to the study of resonances of self-adjoint Schroedinger operators which
have dilation analytic potentials. In spite of the fact that such
eigenfunctions can have surprisingly complicated structures with multiple local
maxima, we show that a suitable adaptation of the JWKB method is able to
provide accurate lobal approximations to them.
