Calculation of eigenvalues of a strongly chaotic system using Gaussian wavepacket dynamics

We apply the approximate dynamics derived from the Gaussian time-dependent variational principle to the Hamiltonian $\hat H= {1/2}(\hat p_x ^2+ \hat p_y ^2)+ {1/2}\hat x^2\hat y^2$, which is strongly chaotic in the classical limit. We are able to calculate, essentially analytically, low-lying eigenvalues for this system. These approximate eigenvalues agree within a few percent with the numerical results. We believe that this is the first example of the use of TDVP dynamics to compute individual eigenvalues in a non-trivial system and one of the few such computations in a chaotic system by any method. There is a short self-contained discussion on the validity of Gaussian approximations in the paper.