The pairing Hamiltonian constitutes an important approximation in many-body systems; it is exactly soluble and quantum integrable. On the other hand, the continuum single-particle level density (CSPLD) contains information about the continuum energy spectrum. The question of whether one can use the Hamiltonian with constant pairing strength for correlations in the continuum is still unanswered. In this paper we generalize the Richardson exact solution for the pairing Hamiltonian including correlations in the continuum. The resonant and nonresonant continua are included through the CSPLD. The resonant correlations are made explicit by using the Cauchy theorem. Low-lying states with seniority 0 and 2 are calculated for the even carbon isotopes. We conclude that energy levels can indeed be calculated with constant pairing in the continuum using the CSPLD. It is found that the nucleus ${}^{24}$C is unbound. The real and complex energy representations of the continuum is developed and their differences are shown. The trajectory of the pair energies in the continuum for the nucleus ${}^{28}$C is shown.

• Received 10 February 2012

DOI:https://doi.org/10.1103/PhysRevC.85.064309