Analysis of previous microscopic calculations for the second $0^{+}$ state in ${}^{12}\mathrm{C}$ in terms of $3-\alpha$ particle Bose-condensed state

The wave function of the second $0^{+}$ state of ${}^{12}\mathrm{C},$ which was obtained a long time ago by solving the microscopic $3\alpha$ problem, is shown to be almost completely equivalent to the wave function of the $3\alpha$ condensed state, which has been proposed recently by the present authors. This equivalence of the wave functions is shown to hold in two cases where different effective two-nucleon forces are adopted. This finding gives strong support for interpreting the second $0^{+}$ state of ${}^{12}\mathrm{C},$ which is the key state for the synthesis of ${}^{12}\mathrm{C}$ in stars (“Hoyle” state), and which is one of the typical mysterious $0^{+}$ states in light nuclei, as a gaslike structure of three $\alpha$ particles, Bose condensed into an identical s-wave function.