  In this article we apply results of \cite{W} on the twisted Mellin transform
to problems in toric geometry. In particular we use these results to describe
the asymptotics of probability densities associated with the monomial
eigenstates, $z^k$, $k \in \ZZ^d$, in Bargmann space and prove an "upstairs"
version of the spectral density theorem of \cite{BGU}. We also obtain for the
$z^k$'s, "upstairs" versions of the results of \cite{STZ} on distribution laws
for eigenstates on toric varieties.
