  The possibility to resolve narrow structures in reaction cross sections in
calculations with the Lorentz integral transform (LIT) method is studied. To
this end we consider a fictitious two-nucleon problem with a low-lying and
narrow resonance in the $^3P_1$ nucleon-nucleon partial wave and calculate the
corresponding ``deuteron photoabsorption cross section''. In the LIT method the
use of continuum wave functions is avoided and one works instead with a
localized function \tilde\Psi. In this case study it is investigated how far
into the asymptotic region \tilde\Psi has to be determined in order to obtain a
precise resolution of the artificially introduced E1 resonance. Comparing with
the results of a conventional calculation with explicit neutron-proton
continuum wave functions it is shown that the LIT approach leads to an
excellent reproduction of the cross section in the resonance region and of
further finer cross section details at higher energies. To this end, however,
for \tilde\Psi one has to take into account two-nucleon distances up to at
least 30 fm.
