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

The wave function of the second $0^+$ state of $^{12}$C which was obtained 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 $^{12}$C which is the key state for the synthesis of $^{12}$C in stars ('Hoyle' state), and which is one of the typical mysterious $0^+$ states in light nuclei, as a gas-like structure of three $\alpha$ particles, Bose-condensed into an identical s-wave function.