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In this new era of radioactive beam facilities, the discovery of novel modesof excitation in nuclei far away from stability represents an area of intenseresearch activity. In addition, these modes of excitation appear to besensitive to the uncertain density dependence of the symmetry energy. We studythe emergence, evolution, and nature of both the soft and giant isoscalarmonopole modes as a function of neutron excess in three unstable Nickelisotopes: 56Ni, 68Ni, and 78Ni. The distribution of isoscalar monopole strengthis computed in a relativistic random-phase approximation using severalaccurately calibrated effective interactions. In particular, a non-spectralGreen's function approach is adopted that allows for an exact treatment of thecontinuum without any reliance on discretization. The discretization of thecontinuum is neither required nor admitted. In the case of 56Ni, the lack oflow-energy strength results in a direct correlation between the centroid energyof the giant monopole resonance and the incompressibility coefficient ofsymmetric nuclear matter. In contrast, the large neutron excess in both 68Niand 78Ni generates a significant, yet relatively featureless, amount oflow-energy strength that is driven by transitions into the continuum. Moreover,the evolution of monopole strength with neutron excess displays sensitivity tothe density dependence of the symmetry energy. Our results suggest that futuremeasurements of the distribution of isoscalar monopole strength at radioactivebeam facilities using a very long chain of both stable and unstable isotopescould place important constraints on the equation of state of neutron-richmatter and ultimately on the properties of neutron stars. However, given thenature of the low-energy monopole excitations, a proper treatment of thecontinuum is essential.

Nuclear breathing mode in neutron-rich Nickel isotopes: sensitivity to the symmetry energy and the role of the continuum