Physics Letters B 321 (1994) 303-306 North-Holland PHYSICS LETTERS B Transformation to pseudo-spin-symmetry of a deformed Nilsson hamiltonian O. Castafios, V. Vehizquez A. Instituto de Ciencias Nucleares, UNAM, Apdo. Postal 70-543, 04510 M~xico, D.F., Mexico P.O. Hess 1 Institut fiir Theoretische Physik, Justus-Liebig Universitllt, Heinrich-Buff-Ring 16, D-35392 Giessen, Germany and J.G. Hirsch Departamento de Fisica, Centro de Investigaci6n y Estudios Avanzados del IPN, Apdo. Postal 14- 740. 07000 M~xico, D.F., Mexico Received 12 November 1993 Editor: G.F. Bertsch The explicit form of the unitary operator connecting the normal parity Nilsson states, of a shell N, with the full set of eigenstates of a pseudodeformed oscillator is given. It is shown that after the transformation pseudo-spin partners are nearly degenerate. The transformation differs from the one of the spherical case indicating a dissimilar definition of normal and unique parity spaces. The division of the shell model space into normal and unique as proposed in the asymptotic limit may be useful in applications to superdeformation phenomena. In the last years interest has increased in the appli- cation of the pseudo-SU (3) (S'~'~)) coupling scheme [ 1,2 ] for discussing and studying a consid- erable variety of phenomena, e.g. to describe the low lying collective states in heavy nuclei [ 3 ], or as an alternative formalism to explain the double beta de- cay [ 4 ], and possibly as a fundamental symmetry as- sociated to the superdeformed bands [ 5 ]. This is due to the fact that within the SU(3) model the spin- orbit interaction, which destroys normal oscillator symmetry, is taken into account effectively and thus allows the use of an algebraic model to describe the low-lying levels of heavy deformed nuclei. Recently it was shown that the harmonic oscillator eigenstates ~r Supported in part by projects CONACYT 1570-E9208, UNAM-DGAPA IN-103091. Permanent address: Instituto de Ciencias Nucleares, UNAM, 04510 M6xico, DF, Mexico. Elsevier Science B.V. SSDI 0370-2693 (93)E1505-R of a particle with spin corresponding to a given num- ber of phonons N, disregarding those with j=N+ ½, which in heavy nuclei are pushed down among the levels of the (N-1 )th shell and are called unique parity states, can be mapped onto the eigenstates of a pseudo oscillator via a unitary transformation Uo as follows [ 6,7 ]: Uo IN(l, s)jm) = I_N(r, g ) f ~ ) , ( 1 ) where ?7=N-l, g=s, f=j, r~=m, (2a) and 7"=l+ 1 according to whetherj=l+ ½. (2b) For example, for N= 4, where in spectroscopic nota- tion we have the levels Sl/2, d3/2, d5/2, g7/2, and g9/2, we disregard g9/2, and map the remaining levels onto the N= 3 states, 303