Quantum Bertrand systems

In this thesis we have analyzed a class of maximally superintegrable systems with radial symmetry on Non-Euclidean manifolds following two distinct footpaths: to begin with we have introduced a class of 1-dimensional exactly solvable quantum systems known as shape invariant systems. This class of systems can be solved by applying algebraic techniques and their eigenfunctions can be described in terms of orthogonal polynomials; afterwards we changed the scale of the physical problems analyzed (from quantum to classical mechanics): from a classical point of view to solve exactly a system means to know the trajectory in the phase space; to this aim we introduced a class of systems whose trajectory can be determined without solving explicitly the equations of motion, namely the Maximally Superintegrable (M.S.) systems. This preliminary analysis introduces the possibility to establish a connection between the classical M.S. systems on non-Euclidean manifolds and quantum exactly solvable systems. Through this connection has been possible to obtain a quantization recipe for non- Euclidean systems which preserves the exact solvability and the dynamical symmetries of the classical version, this is highly non trivial because of the so-called "ordering problem" that always arises in presence of a non at space. The main e ect of this superintegrable quantization can be recognized through the presence of the "accidental" degeneracy in the spectrum which is a characteristic feature shared by all maximally superintegrable quantum systems. 1