Bertrand systems on spaces of constant sectional curvature. The action-angle analysis. Classical, quasi-classical and quantum problems (Q2825593)

The present large paper deals with completely degenerate Bertrand-like systems on three-dimensional real space forms. A main tool is the analysis of the involved groups, namely \(\mathrm{SU}(2)\), \(\mathrm{SO}(3,\mathbb R)\) and \(\mathrm{SL}(2,\mathbb R)\) with its quotient \(\mathrm{SO}(1,2)\). This type of problems is studied from three points of view: the classical action-angle description, the associated quasi-classical analysis and the quantum version using the Bohr-Sommerfeld quantization rule. An important conclusion is that the classical action-angle case and the quantum mechanical version are superpositions of the flat-space expression.NEWLINENEWLINEFor the entire collection see [Zbl 1317.00020].