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The peculiarity of the Eckart potential problem on ${\bf H}^2_+$ (the uppersheet of the two-sheeted two-dimensional hyperboloid), to preserve the$(2l+1)$-fold degeneracy of the states typical for the geodesic motion there,is usually explained in casting the respective Hamiltonian in terms of theCasimir invariant of an so(2,1) algebra, referred to as potential algebra. Ingeneral, there are many possible similarity transformations of the symmetryalgebras of the free motions on curved surfaces towards potential algebras,which are not all necessarily unitary. In the literature, a transformation ofthe symmetry algebra of the geodesic motion on ${\bf H}^2_+$ towards thepotential algebra of Eckart's Hamiltonian has been constructed for the primepurpose to prove that the Eckart interaction belongs to the class of Natanzonpotentials. We here take a different path and search for a transformation whichconnects the $(2l+1)$ dimensional representation space of the pseudo-rotationalso(2,1) algebra, spanned by the rank-$l$ pseudo-spherical harmonics, to therepresentation space of equal dimension of the potential algebra and find atransformation of the scaling type. Our case is that in so doing one isproducing a deformed isometry copy to ${\bf H}^2_+$ such that the free motionon the copy is equivalent to a motion on ${\bf H}^2_+$, perturbed by a $\coth$interaction. In this way, we link the so(2,1) potential algebra concept of theEckart Hamiltonian to a subtle type of pseudo-rotational symmetry breakingthrough ${\bf H}^2_+$ metric deformation.

Breaking Pseudo-Rotational Symmetry through ${\bf H}^2_+$ Metric Deformation in the Eckart Potential Problem