Exact solvability of superintegrable Benenti systems

Completely integrable systems in classical mechanics are well known to be of great interest for both theory and applications (see e.g. [1, 2] and references therein). Indeed, the possibility of analytical description of the corresponding dynamics, enables us to uncover important physical properties of the corresponding systems. The prime examples of this are the Kepler laws and numerous physical models based on (superposition of) harmonic oscillators. Interestingly, a number of physically relevant exactly solvable models (e.g., the Coulomb problem for the hydrogen atom and the multidimensional harmonic oscillator, to name just a few) are maximally superintegrable rather than just completely integrable. In general, a Hamiltonian dynamical system on a 2n-dimensional phase space with a Hamiltonian H is maximally superintegrable, if it possesses the maximal possible number, 2n−1, of functionally independent, globally defined integrals of motion (contrast this with n commuting integrals of motion for completely integrable systems); see e.g. [3, 4] and references therein for further details. For the sake of brevity we shall also say that the Hamiltonian in question, H , is maximally superintegrable. Quantizing a generic completely integrable classical system and solving the resulting Schrödinger equation may often represent a nontrivial problem. The latter is usually considerably alleviated for the maximally superintegrable systems (sometimes to the extent of reducing the determination of energy spectrum to a purely algebraic problem, as is e.g. the case for the nonrelativistic hydrogen atom and the multidimensional harmonic oscillator) because of the presence of additional integrals of motion [5].