It is proved that the potentials of the form β 2n (with n being integer) provide a "bridge" between the U(5) symmetry of the Bohr Hamiltonian with a harmonic oscillator potential (occuring for n = 1) and the E(5) model of Iachello (Bohr Hamiltonian with an infinite well potential, materialized for n → ∞). Parameter-free (up to overall scale factors) predictions for spectra and B(E2) transition rates are given for the potentials β 4 , β 6 , β 8 , corresponding to R 4 = E(4)/E(2) ratios of 2.093, 2.135, 2.157 respectively, compared to the R 4 ratios 2.000 of U(5) and 2.199 of E(5). Hints about nuclei showing this behaviour, as well as about potentials "bridging" the E(5) symmetry with O(6) are briefly discussed. A note about the appearance of Bessel functions in the framework of E(n) symmetries is given as a by-product.

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The q-analogue of the SU(2)-algebra, namely SU,(2), is applied to the description of the energy spectra of the deformed even-even nuclei. The theoretical results are compared with the available experimental data.

Starting from the original collective Hamiltonian of Bohr and separating the β and γ variables as in the X(5) model of Iachello, an exactly soluble model corresponding to a harmonic oscillator potential in the β-variable (to be called X(5)-β 2 ) is constructed. Furthermore, it is proved that the potentials of the form β 2n (with n being integer) provide a "bridge" between this new X(5)-β 2 model (occuring for n = 1) and the X(5) model (corresponding to an infinite well potential in the β-variable, materialized for n → ∞). Parameter-free (up to overall scale factors) predictions for spectra and B(E2) transition rates are given for the potentials β 2 , β 4 , β 6 , β 8 , corresponding to R 4 = E(4)/E(2) ratios of 2.646, 2.769, 2.824, and 2.852 respectively, compared to the R 4 ratios of 2.000 for U(5) and 2.904 for X(5). Hints about nuclei showing this behaviour, as well as about potentials "bridging" the X(5) symmetry with SU(3) are briefly discussed.PACS numbers: 21.60.Ev, 21.60.Fw, 21.10.Re 1. Introduction Models providing parameter-independent predictions for nuclear spectra and electromagnetic transition rates serve as useful benchmarks in nuclear theory. The recently introduced E(5) [1] and X(5) [2] models belong to this category, since their predictions for nuclear spectra (normalized to the excitation energy of the first excited state) and B(E2) transition rates (normalized to the B(E2) transition rate connecting the first excited state to the ground state) do not contain any free parameters. The E(5) model appears to be related to a phase transition from U(5) (vibrational) to O(6) (γ-unstable) nuclei [1], while X(5) is related to a phase transition from U(5) (vibrational) to SU(3) (prolate deformed) nuclei [2]. Both models originate (under certain simplifying assumptions) from the Bohr collective Hamiltonian [3], which is known to possess the U(5) symmetry of the five-dimensional (5-D) harmonic oscillator [4].In the present paper we study a sequence of potentials lying between the U(5) symmetry of the Bohr Hamiltonian and the X(5) model. The potentials, which are of the form 1

Davidson potentials of the form β 2 + β 4 0 /β 2 , when used in the original Bohr Hamiltonian for γ-independent potentials bridge the U(5) and O(6) symmetries. Using a variational procedure, we determine for each value of angular momentum L the value of β 0 at which the derivative of the energy ratio R L = E(L)/E(2) with respect to β 0 has a sharp maximum, the collection of R L values at these points forming a band which practically coincides with the ground state band of the E(5) model, corresponding to the critical point in the shape phase transition from U(5) to O(6). The same potentials, when used in the Bohr Hamiltonian after separating variables as in the X(5) model, bridge the U(5) and SU(3) symmetries, the same variational procedure leading to a band which practically coincides with the ground state band of the X(5) model, corresponding to the critical point of the U(5) to SU(3) shape phase transition. A new derivation of the Holmberg-Lipas formula for nuclear energy spectra is obtained as a by-product.

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