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The pairing interaction is one of the most important contribution of theresidual interaction and then, it is of major importance for the study ofmany-body systems. One can get solutions of the pairing Hamiltonian throughtthe Bardeen-Cooper-Schieffer (BCS) or the Lipkin-Nogami (LN) approximationsbut, the pairing Hamiltonian admit exact solution worked out by Richardson.Nuclei far away from the stability line have important correlations with thecontinuum part of the energy spectrum, due that the Fermi level is very closeto the contiuum thershold. The correlations with the continuum can be includedin the many-body description through the complex energy states, called Gamowstates. In this work we compare the approximates and exact solutions of thepairing Hamiltonian in real and complex-energy representations. In theapplication of this formulation to the symmetric Lipkin model, we found thatthe LN solution is in a good agreement with the exact one; besides, theextension of the BCS solution to the complex energy plane gives solution evenfor strength below the critical one, which is purely imaginary.

Comparison between the Lipkin-Nogami and Richardson solutions with complex single particle energy in the Lipkin's model