  In this paper, building on a previous analysis [1] of exact diagonalization
of the space-discretized evolution operator for the study of properties of
non-relativistic quantum systems, we present a substantial improvement to this
method. We apply recently introduced effective action approach for obtaining
short-time expansion of the propagator up to very high orders to calculate
matrix elements of space-discretized evolution operator. This improves by many
orders of magnitude previously used approximations for discretized matrix
elements and allows us to numerically obtain large numbers of accurate energy
eigenvalues and eigenstates using numerical diagonalization. We illustrate this
approach on several one and two-dimensional models. The quality of numerically
calculated higher order eigenstates is assessed by comparison with
semiclassical cumulative density of states.
