From E(5) to shape coexistence: a catastrophe-theoretic phase diagram and Maxwell critical curve for the γ -unstable Bohr model

Abstract A comprehensive theoretical investigation of shape coexistence and quantum phase transitions (QPTs) within the γ -unstable Bohr Hamiltonian framework is presented. The model employs a flexible sextic polynomial potential, β 2 − a β 4 + b β 6 , capable of producing spherical, deformed, and coexisting configurations. High-accuracy solutions of the radial Schrödinger equation were obtained via a Chebyshev spectral collocation method, enabling computation of spectroscopic observables—energy ratios and B(E2) transition strengths—with machine precision. The topology of the effective potential was characterized using catastrophe-theoretic methods: analytic expressions for spinodal boundaries delimiting the coexistence wedge and for the Maxwell set of equal-depth minima were derived, revealing a cusp that organizes the global phase diagram and clarifies proximity to critical-point behavior. Resulting wave-function morphologies and tunneling patterns were shown to produce systematic interband transition trends, providing diagnostic fingerprints that distinguish first-order from second-order QPTs and situating the E(5) limit as a cusp-adjacent case. An inverse-problem protocol including an L ( L + 1 ) term to lift SO(5) degeneracy was formulated. Two complementary optimization strategies were implemented: a soft-constraint search over the coexistence region and a hard-constraint fit confined to the Maxwell curve, enabling quantitative assessment of a nucleus’s distance to criticality and degree of shape mixing. The model reproduces low-lying spectra and shape-mixing evolution in 72 Kr, 186 Hg, and 108 Pd nuclei with good agreement to experiment.