Eigenstates and Spectral Projection for Quantized Baker’s Map

Abstract

We extend the approach from Shou (Ann Henri Poincaré 24:2833–2875, 2023) to prove windowed spectral projection estimates and a generalized Weyl law for the (Weyl) quantized baker’s map on the torus. The spectral window is allowed to shrink in the semiclassical (large dimension) limit. As a consequence, we obtain a strengthening of the quantum ergodic theorem from Degli Esposti et al. (Commun Math Phys 263(2):325–352, 2006) to hold in shrinking spectral windows, a Weyl law on uniform spreading of eigenvalues, and statistics of random quasimodes. Using similar techniques, we also investigate random eigenbases of a different (non-Weyl) quantization, the Walsh-quantized baker’s map, which has high degeneracies in its spectrum. For such random eigenbases, we prove that Gaussian eigenstate statistics and QUE hold with high probability in the semiclassical limit.