The Berry-Keating operator on a lattice

We construct and study a version of the Berry-Keating operator corresponding to a classical Hamiltonian on a compact 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 only a specific combination of the limits leads to a logarithmic mean spectral density as it was anticipated by Berry and Keating.