Confocal parabolic billiard with gravitational potential: Classical and quantum description

We investigate the classical and quantum dynamics of a particle trapped in a gravitational confocal parabolic billiard. Characterizing the equi-momentum surfaces and the Poincar & eacute; phase maps reveals four different kinds of motion the particle can exhibit. The analytical expressions of the characteristic equations for getting periodic orbits and their periods were derived and validated numerically. A notable finding is the possibility of having degenerate periodic trajectories with the same energy but different second constants of motion and caustics. Eigenstates of the particle with definite values of the constants of motion can be associated with classical orbits with the same pair of constants. To make this correspondence we use periodic orbits.