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Perturbing free motions on hyper spheres without degeneracy lift

We consider quantum motion on S3 perturbed by the trigonometric Scarf potential (Scarf I) with one internal quantized dimensionless parameter, l, the ordinary orbital angular momentum value, and another, continuous parameter, b, through which an external scale is introduced. We argue that a loss of the geometric hyper-spherical so(4) symmetry of the free motion occurs that leaves intact the unperturbed hydrogen-like degeneracy patterns characterizing the spectrum under discussion. The argument is based on the observation that the expansions of the Scarf I wave functions for fixed l-values in the basis of properly identified $so(4)$ representation functions are power series in the perturbation parameter, b, in which carrier spaces of dimensionality (K+1)^2 with K varying as K\in [l, N-1], and N being the principal quantum number of the Scarf I potential problem, contribute up to the order O(b^(N-1-K)). Nonetheless, the degeneracy patterns can still be interpreted as a consequence of an effective so(4) symmetry, i.e. a symmetry realized at the level of the dynamic of the system, in so far as from the perspective of the eigenvalue problem, the Scarf I results are equivalently obtained from a Hamiltonian with matrix elements of polynomials in a properly identified so(4) Casimir operator. The scheme applies to any dimension, d.