The Mellin transform and spectral properties of toric varieties

TL;DR

This paper explores the spectral properties of toric varieties using the Mellin transform, providing asymptotic analysis of probability densities and distribution laws for eigenstates in Bargmann space.

Contribution

It applies Mellin transform techniques to toric geometry, deriving new asymptotic results and spectral density theorems for eigenstates on toric varieties.

Findings

Asymptotic descriptions of probability densities for monomial eigenstates.

An 'upstairs' spectral density theorem for toric varieties.

Distribution laws for eigenstates on toric varieties.

Abstract

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.