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 tildePsi. In this case study it is investigated how far into the asymptotic region tildePsi 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 tildePsi one has to take into account two-nucleon distances up to at least 30 fm.
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