Breaking pseudo-rotational symmetry through $\mathbf H^2_+$ metric deformation in the Eckart potential problem

The peculiarity of the Eckart potential problem on $\mathbf H^2_+$ (the upper sheet 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 the Casimir invariant of an so$(2,1)$ algebra, referred to as potential algebra. In general, there are many possible similarity transformations of the symmetry algebras of the free motions on curved surfaces towards potential algebras, which are not all necessarily unitary. In the literature, a transformation of the symmetry algebra of the geodesic motion on $\mathbf H_+^2$ towards the potential algebra of Eckart's Hamiltonian has been constructed for the prime purpose to prove that the Eckart interaction belongs to the class of Natanzon potentials. We here take a different path and search for a transformation which connects the $(2l+1)$ dimensional representation space of the pseudo-rotational so$(2,1)$ algebra, spanned by the rank-$l$ pseudo-spherical harmonics, to the representation space of equal dimension of the potential algebra and find a transformation of the scaling type. Our case is that in so doing one is producing a deformed isometry copy to $\mathbf H^2_+$ such that the free motion on the copy is equivalent to a motion on $\mathbf H^2_+$, perturbed by a $\coth$ interaction. In this way, we link the so$(2,1)$ potential algebra concept of the Eckart Hamiltonian to a subtle type of pseudo-rotational symmetry breaking through $\mathbf H^2_+$ metric deformation. From a technical point of view, the results reported here are obtained by virtue of certain nonlinear finite expansions of Jacobi polynomials into pseudo-spherical harmonics. In due places, the pseudo-rotational case is paralleled by its so(3) compact analogue, the cotangent perturbed motion on S$^2$. We expect awareness of different so$(2,1)$/so$(3)$ isometry copies to benefit simulation studies on curved manifolds of many-body systems.