  Proton-rich nuclei possess unique properties in the nuclear chart. Due to the
presence of both continuum coupling and Coulomb interaction, phenomena such as
halos, Thomas-Ehrman shift, and proton emissions can occur. Experimental data
are difficult to be obtained therein, so that theoretical calculations are
needed to understand nuclei at drip-lines and guide experimentalists for that
matter. In particular, the $^{16}$Ne and $^{18}$Mg isotopes are supposed to be
one-proton and/or two-proton emitting nuclei, but associated experimental data
are either incomplete or even unavailable. Consequently, we performed Gamow
shell model calculations of carbon isotones bearing $A=15\text{-}18$.
Isospin-symmetry breaking occurring in carbon isotones and isotopes is also
discussed. It is hereby shown that the mixed effects of continuum coupling and
Coulomb interaction at drip-lines generate complex patterns in isospin
multiplets. Added to that, it is possible to determine the one-proton and
two-proton widths of $^{16}$Ne and $^{18}$Mg. Obtained decay patterns are in
agreement with those obtained in previous experimental and theoretical works.
Moreover, up to the knowledge of authors, this is the first theoretical
calculation of binding energy and partial decay widths of $^{18}$Mg in a
configuration interaction picture.
