Sequence of potentials interpolating between the U(5) and E(5) symmetries

It is proved that the potentials of the form beta(2n) (with n being integer) provide a "bridge" between the U(5) symmetry of the Bohr Hamiltonian with a harmonic oscillator potential (occurring for n=1) and the E(5) model of Iachello (Bohr Hamiltonian with an infinite well potential, materialized for n-->infinity). Parameter-free (up to overall scale factors) predictions for spectra and B(E2) transition rates are given for the potentials beta(4), beta(6), and beta(8), corresponding to R-4=E(4)/E(2) ratios of 2.093, 2.135, and 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 behavior, 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.