Deformed harmonic oscillators for metal clusters: Analytic properties and supershells

The analytic properties of Nilsson’s modified oscillator, which was first introduced in nuclear structure, and of the recently introduced, based on quantum algebraic techniques, three-dimensional q-deformed harmonic oscillator (3D $q-\mathrm{HO})$ with ${\mathrm{u}}_q\mathrm{}\left(3\right)\mathrm{}\supset {\mathrm{so}}_q\mathrm{}\left(3\right)\mathrm{}$ symmetry, which is known to reproduce correctly in terms of only one parameter the magic numbers of alkali clusters up to 1500 (the expected limit of validity for theories based on the filling of electronic shells), are considered. Exact expressions for the total energy of closed shells are determined and compared among them. Furthermore, the systematics of the appearance of supershells in the spectra of the two oscillators is considered, showing that the 3D $q-\mathrm{HO}$ correctly predicts the first supershell closure in alkali clusters without use of any extra parameter.