  Wigner functions play a central role in the phase space formulation of
quantum mechanics. Although closely related to classical Liouville densities,
Wigner functions are not positive definite and may take negative values on
subregions of phase space. We investigate the accumulation of these negative
values by studying bounds on the integral of an arbitrary Wigner function over
noncompact subregions of the phase plane with hyperbolic boundaries. We show
using symmetry techniques that this problem reduces to computing the bounds on
the spectrum associated with an exactly-solvable eigenvalue problem and that
the bounds differ from those on classical Liouville distributions. In
particular, we show that the total ``quasiprobability'' on such a region can be
greater than 1 or less than zero.
