  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.
