Background: Weakly bound and unbound nuclei close to particle drip lines are laboratories of new nuclear structure physics at the extremes of neutron/proton excess. The comprehensive description of these systems requires an open quantum system framework that is capable of treating resonant and nonresonant many-body states on equal footing. Purpose: In this work, we construct the minimal complex-energy configuration interaction approach to describe binding energies and spectra of selected 5 $leq$ A $leq$ 11 nuclei. Method: We employ the complex-energy Gamow shell model (GSM) assuming a rigid $^4$He core. The effective Hamiltonian, consisting of a core-nucleon Woods-Saxon potential and a simplified version of the Furutani-Horiuchi-Tamagaki interaction with the mass-dependent scaling, is optimized in the sp space. To diagonalize the Hamiltonian matrix, we employ the Davidson method and the Density Matrix Renormalization Group technique. Results: Our optimized GSM Hamiltonian offers a good reproduction of binding energies and spectra with the root-mean-square (rms) deviation from experiment of 160 keV. Since the model performs well when used to predict known excitations that have not been included in the fit, it can serve as a reliable tool to describe poorly known states. A case in point is our prediction for the pair of unbound mirror nuclei $^{10}$Li-$^{10}$N in which a huge Thomas-Ehrman shift dramatically alters the pattern of low-energy excitations. Conclusion: The new model will enable comprehensive studies of structure and reactions aspects of light drip-line nuclei.
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