  We extend the approach from [arXiv:2110.15301] 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. As a
consequence, we obtain a strengthening of the quantum ergodic theorem from
[arXiv:math-ph/0412058] to hold in shrinking spectral windows, a Weyl law on
uniform spreading of eigenvalues, and statistics of random band-limited waves.
Using similar techniques, we also investigate generic random eigenbases of a
different (non-Weyl) quantization, the Walsh-quantized $D$-baker map, which has
high degeneracies in its spectrum. For generic random eigenbases, we prove
gaussian eigenstate statistics and QUE with high probability in the
semiclassical limit.
