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Lowest energy spectrum of the $^{12}$C nucleus is analyzed in the 3$\alpha$cluster model with a deep $\alpha\alpha$-potential of Buck, Friedrich andWheatley with Pauli forbidden states in the $S$ and $D$ waves. The directorthogonalization method is applied for the elimination of the 3$\alpha$-Pauliforbidden states. The effects of possible first order quantum phase transitionare shown in the lowest $^{12}$C($0_1^+)$ and $^{12}$C($2_1^+)$ states fromweakly bound phase to a deep phase. The ground and lowest $2^+$ states of the$^{12}$C nucleus in the deep phase are created by the critical eigen states ofthe Pauli projector for the $0^+$ and $2^+$ three-alpha functional spaces,respectively.

Peculiarity of the $^{12}$C$(0^+)$ and $^{12}$C$(2^+)$ energy spectrum in a 3$\alpha$ model