The irreducible representation labels 2 and /t of the SU(3) shell model are related to the shape variables/~ and 7 of the collective model by invoking a linear mapping between eigenvalues of invariant operators of the two theories. All but one parameter of the theory is fixed if the shell-model result is required to reproduce the collective-model geometry. And for one special value of the remaining free parameter there is a simple linear relationship between the eigenvalues, 2,, of the quadrupole matrix of the collective model and the SU(3) representation labels:The correspondence between hamiltonians that describe rotations in each theory is also given. Results are shown for two cases, 24Mg and 168Er, to demonstrate that the simplest mapping yields excellent results for both energies and transition rates. For 2 and/or # large, the (/3, 7)~--~(2,/~) correspondence introduced here reduces to the symplectic shell-model result.

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A new form to construct complex superpotentials that produce real energy spectra in supersymmetric quantum mechanics is presented. This is based on the relation between the nonlinear Ermakov equation and a second order differential equation of the Schrödinger type. The superpotentials so constructed are characterized by the Ermakov parameters in such a way that they are always complex-valued and lead to non-Hermitian Hamiltonians with real spectra, whose eigenfunctions form a bi-orthogonal system. As applications we present new complex supersymmetric partners of the free particle that are PT -symmetric and can be either periodic or regular (of the Pöschl-Teller form). A new family of complex oscillators with real frequencies that have the energies of the harmonic oscillator plus an additional real eigenvalue is introduced.

The Dicke Hamiltonian describes the simplest quantum system with atoms interacting with photons: N two-level atoms inside a perfectly reflecting cavity, which allows only one electromagnetic mode. It has also been successfully employed to describe superconducting circuits that behave as artificial atoms coupled to a resonator. The system exhibits a transition to a superradiant phase at zero temperature. When the interaction strength reaches its critical value, both the number of photons and atoms in excited states in the cavity, together with their fluctuations, exhibit a sudden increase from zero. By employing symmetry-adapted coherent states, it is shown that these properties scale with the number of atoms, their reported divergences at the critical point represent the limit when this number goes to infinity, and, in this limit, they remain divergent in the superradiant phase. Analytical expressions are presented for all observables of interest, for any number of atoms. Comparisons with exact numerical solutions strongly support the results.