The Mellin Transform and Spectral Properties of Toric Varieties

In this paper we apply the results of 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 \mathbb{Z}^{d} $ , in Bargmann space and prove an “upstairs” version of the spectral density theorem of BGU]. We also obtain for the z ^k ’s, “upstairs” versions of the results of STZ] on distribution laws for eigenstates on toric varieties.