Propagation of the nuclear mean-field uncertainties with increasing distance from the parameter adjustment zone: Applications to superheavy nuclei

We combine the framework of the inverse problem theory and the Monte Carlo approach to formulate exact mathematical models that enable estimates of the uncertainty distributions for modeling predictions. We illustrate and discuss in particular what we refer to as the ``NO GO property.'' When the uncertainties of data constituting the input to the parameter adjustment procedures exceed certain critical value(s), even an exact modeling looses its stochastic reliability; its further use may provide ``acceptably looking'' rms deviations in the fitting zone with very likely meaningless, because they are unstable, predictions outside of it. We examine confidence intervals for intraneous (inside of the adjustment zone) and extraneous (outside of the adjustment zone) predictions and we demonstrate that ``satisfactory'' rms deviations in the intraneous modeling regime offer generally null certitude about the quality of extraneous predictions. Even though not entirely unknown, this property requires strong emphasizing since ignoring it has lead to misleading conclusions and confusing messages in the literature. We generalize our considerations to the realistic nuclear mean-field simulations of the properties of the nucleonic mean-field energies in spherical nuclei. We predict quantitatively the deterioration with increasing mass of the nucleonic-energy confidence intervals in superheavy nuclei. We show a strong dependence of those confidence intervals on the quantum characteristic of nucleonic states and provide detailed illustrations. In particular we demonstrate that, in the realistic predictions for the superheavy nuclei with the phenomenological Woods-Saxon Hamiltonian for up to $Z\ensuremath{\approx}114$ or so, one obtains relatively stable predictions of the single-particle spectra with $N\ensuremath{\le}180$, while approaching the NO GO zone of this model for further increasing neutron numbers. Thus the main area of today's interest within the instrumental reach for the superheavy nuclei studies remains, according to these estimates for the Woods-Saxon modeling, within the stability zone.