The covering SU(3) group over anisotropic harmonic oscillators

We propose new non-linear boson transformation by which all the anisotropic oscillator states can be embedded in the SU(3) bases. We start from the oscillator Hamiltonian without spin- orbit interaction, and suppose that three oscillator frequencies have an integral rational ratio $a:b:c$. In order to construct a SU(3)-invariant expression, we express the harmonic oscillator boson operator $c_k$ ($k=x,y,z$), in terms of a $m$-fold product of new bosons $s_m$ ($m=a,b,c$), by requiring $s_{m}^{\dag}s_m=mc_{k}^{\dag}c_k$. The general form of the new bosons $s_m$, for any positive integer $m$, is given by $c_{k}=[ m \prod_{r=1}^{m-1}({\hat n}_m +r) ]^{-1/2}(s_{m}) ^m$, with ${\hat n}_{m}=s_m^{\dag}s_m$. Applying the analogy of Elliott's group operators, we obtain a similar set of group operators from new bosons $s_a$, $s_b$ and $s_c$, i.e., ${\tilde Q}_q$ for $q=0, \pm 1$ and $\pm 2$, and ${\tilde \ell}_k$ for $k=a,b$ and $c$. Then, the commutation relations among these 8 operators are closed, and they commute with $H$. Together with Casimir operator and two operators which have diagonal form in number operators, i.e., ${\tilde Q}_{0}$, and ${\tilde Q}_{2}+ {\tilde Q}_{-2}$, we can classify the single-particle states in $N_{\rm sh}$, and find the new magic numbers for the triaxially deformed field.