Pointwise Weyl Law for Graphs from Quantized Interval Maps

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 Pakonski et al. (J Phys A 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.