The Mellin transform and spectral properties of toric varieties

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, $z^k$, $k\in {\text{ZZ}}^d$, 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.