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

Lowest energy spectrum of the $^{12}$C nucleus is analyzed in the 3$α$ cluster model with a deep $αα$-potential of Buck, Friedrich and Wheatley with Pauli forbidden states in the $S$ and $D$ waves. The direct orthogonalization method is applied for the elimination of the 3$α$-Pauli forbidden states. The effects of possible first order quantum phase transition are shown in the lowest $^{12}$C($0_1^+)$ and $^{12}$C($2_1^+)$ states from weakly 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 of the Pauli projector for the $0^+$ and $2^+$ three-alpha functional spaces, respectively.