Basis truncation, statistical errors, and systematic uncertainties in relativistic approaches to nuclear response

Although there exists a clear and, in principle, exact theoretical formulation for the equation of motion for the response of a correlated fermionic system, its numerical implementations for atomic nuclei require feasible approximations. One of the widely accepted approximations is a truncated harmonic oscillator (HO) basis, whose wave functions are used to expand the solutions obtained with realistic interactions. In this work, we extend previously employed HO basis truncated at $N_F$ = 20 fermionic shells to $N_F$ = 50 and perform a systematic study of the effects of such basis increase on nuclear resonances. The relativistic random phase approximation (RRPA) and its extension by the particle-vibration coupling dubbed as relativistic time-blocking approximation (RTBA) are applied to the description of the monopole, dipole, quadrupole, and octupole resonances in $^{48}$Ca, $^{78}$Ni, and $^{132}$Sn, and the RRPA studies are extended to $^{70}$Ca and $^{208}$Pb. A considerable sensitivity of the strength distributions to the HO basis size is found, especially for low-spin resonances in the light neutron-rich nuclei. The effects of the HO basis extension to $N_F$ = 50 are analyzed and linked to the involvement of proton and neutron continuum states and proton quasi-bound states in the strength formation. The obtained results point to the importance of the HO basis completeness and continuum effects in the nuclear response calculations and evaluation of the associated parameters of the nuclear equation of state. Statistical errors and systematic uncertainties in the RRPA strength functions are analyzed. They are found to be substantial for the monopole response, but significantly smaller for the dipole, quadrupole, and octupole ones. Neither of them shows a pronounced mass dependence, and statistical errors are generally smaller than systematic uncertainties.