The Berry-Keating operator on a lattice

TL;DR

This paper introduces a lattice-based Berry-Keating operator as a Weyl quantization of an inverted harmonic oscillator, analyzing its spectral properties through semiclassical limits on a finite torus.

Contribution

It constructs a lattice version of the Berry-Keating operator with phase space truncation and studies its spectral behavior in various limits.

Findings

Spectral density exhibits a logarithmic growth in the semiclassical limit

Operator defined on a finite, periodic lattice with truncation effects analyzed

Limits lead to spectral properties consistent with Berry and Keating's predictions

Abstract

We construct and study a version of the Berry-Keating operator with a built-in truncation of the phase space, which we choose to be a two-dimensional torus. The operator is a Weyl quantisation of the classical Hamiltonian for an inverted harmonic oscillator, producing a difference operator on a finite, periodic lattice. We investigate the continuum and the infinite-volume limit of our model in conjunction with the semiclassical limit. Using semiclassical methods, we show that a specific combination of the limits leads to a logarithmic mean spectral density as it was anticipated by Berry and Keating.