  We theoretically analyze the excitation energy transfer between two closely
spaced linear molecular J-aggregates, whose excited states are Frenkel
excitons. The aggregate with the higher (lower) exciton band edge energy is
considered as the donor (acceptor). The celebrated theory of F\"orster
resonance energy transfer (FRET), which relates the transfer rate to the
overlap integral of optical spectra, fails in this situation. We point out that
in addition to the well-known fact that the point-dipole approximation breaks
down (enabling energy transfer between optically forbidden states), also the
perturbative treatment of the electronic interactions between donor and
acceptor system, which underlies the F\"orster approach, in general loses its
validity due to overlap of the exciton bands. We therefore propose a
nonperturbative method, in which donor and acceptor bands are mixed and the
energy transfer is described in terms of a phonon-assisted energy relaxation
process between the two new (renormalized) bands. The validity of the
conventional perturbative approach is investigated by comparing to the
nonperturbative one; in general this validity improves for lower temperature
and larger distances (weaker interactions) between the aggregates. We also
demonstrate that the interference between intraband relaxation and energy
transfer renders the proper definition of the transfer rate and its evaluation
from experiment a complicated issue, which involves the initial excitation
condition.
