CONSTANTS OF THE MOTION AND DIAGONALIZATION OF THE HAMILTONIAN FOR SOME SCATTERING PROCESSES

Complex invariants (for SHO modes) or pairs of real invariants (for RHO modes) are obtained for the frequency converter, the parametric amplifier, and for a three-mode system that describes Raman or Brillouin scattering. All possible constants of the motion and the diagonalization of the Hamiltonian follow as a consequence. Under the assumption made in this paper, the Hamiltonian has also been considered by Moshinsky and Winternitz, who have established the existence of its reduction to diagonal form. However, it is not easy to select the correct transformation (from the totality comprising the symplectic group) to reduce a given H. This has been done explicitly showing, among other results, that the constants of the motion are similar (often identical) in the stable (frequency converter) domain and in the unstable (amplifier) domain.