Semi classical chaos in a 2-D exactly solvable system

The Chaotic behavior of the classical trajectories and the level spacing distributions of 2-D Pseudo Hermitian Hamiltonian H =½ (p_x² + p_y²) + ½ (ω_x²x² + ω_y²y²) - igxy, where g is a real parameter are investigated. This Hamiltonian is separable and Lyapunov exponents are obtained analytically. All the trajectories of this system are found to be regular and Hamiltonian is Hermitian when ½(| ω_x² - ω_y²|)≥|g| and trajectories become chaotic and Hamiltonian becomes Pseudo Hermitian when ½(| ω_x² - ω_y²|) < |g|. The level spacing distributions are obtained for both real and complex eigen energies. It was interesting to find out that level spacing distributions of complex norm of the eigen energies which are corresponding to chaotic motion show Poisson like distribution rather than Gaussian ensamble statistics.

doi:10.4038/sljp.v5i0.192  

Sri Lankan Journal of Physics, Vol.5 (2004) 1-9