Pointwise Weyl law for graphs from quantized interval maps

TL;DR

This paper establishes a pointwise Weyl law for quantum graphs derived from interval maps, extending quantum ergodic results and analyzing eigenvector distributions in the semiclassical limit.

Contribution

It introduces a pointwise Weyl law for quantized interval maps, strengthening quantum ergodic theorems and analyzing eigenvector behavior under random perturbations.

Findings

Strengthening of quantum ergodic theorem for these models

Approximate Gaussian distribution of eigenvectors in semiclassical limit

Analysis of the doubling map case

Abstract

We prove an analogue of the pointwise Weyl law for families of unitary matrices obtained from quantization of one-dimensional interval maps. This quantization for interval maps was introduced by Pako\'nski et al. [J. Phys. A: Math. Gen. 34 9303-9317 (2001)] as a model for quantum chaos on graphs. Since we allow shrinking spectral windows in the pointwise Weyl law, as a consequence we obtain for these models a strengthening of the quantum ergodic theorem from Berkolaiko et al. [Commun. Math. Phys. 273 137-159 (2007)], and show in the semiclassical limit that a family of randomly perturbed quantizations has approximately Gaussian eigenvectors. We also examine further the specific case where the interval map is the doubling map.