Eigenstates and spectral projection for quantized baker's map

TL;DR

We analyze quantized Baker maps on the torus using Balazs–Voros Weyl quantization and compare to a Walsh quantization. The core contributions are windowed spectral projection bounds and a generalized Weyl law in shrinking spectral windows, which yield a strengthened quantum ergodic theorem and uniform eigenvalue spreading. The work further develops randomness-based results for quasimodes and shows that Gaussian statistics and quantum ergodicity hold with high probability for random eigenbases in Walsh quantizations. Together these results illuminate fine-scale spectral and eigenfunction behavior in chaotic quantum maps and provide robust probabilistic descriptions of spectral windows approaching the semiclassical limit. The methods hinge on coherent-state techniques, Beurling–Selberg approximations, and careful control of short-time dynamics up to Ehrenfest-like times, extended to a Walsh framework for comparison and contrast.

Abstract

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 in the semiclassical (large dimension) limit. 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 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.

Figures (8)

• Figure 1: Classical baker's map operation.
• Figure 2: The absolute value of the matrix entries of $\widehat{B}_N^k$, for $N=100$ and $k=1,2,3$, plotted on a power scale. For these small powers $k$, the large matrix entries trace out the classical map $x\mapsto 2^kx\;\mathrm{mod}\;1$ (flipped vertically). However, as $k$ becomes larger and reaches the Ehrenfest time, the relation to the classical map begins to collapse, and the matrix entry patterns begin to look fairly random. (By the Ehrenfest time, the graph of $x\mapsto 2^kx\;\mathrm{mod}\;1$ fills up the entire grid.) As $N$ increases, one can allow longer times $k$ before the collapse.
• Figure 3: The "bad set" $B_{J,\delta,\gamma,N}$ is to be excluded due to discontinuities and diffraction effects. The "classical set" $C_{k,N}^W$ is where we expect $\widehat{B}_N^k$ to be (relatively) large. Away from $B_{J,\delta,\gamma,N}$ and $\bigcup_{k=1}^J C_{k,N}^W$, the matrices $\widehat{B}_N^k$, $k=1,\ldots,J$, will have small entries (Theorem \ref{['prop:mpowers']}).
• Figure 4: Example region (shown in blue northwest hatching) used to bound the size of the diagonal set $DA_{J,\delta,\gamma,N}^W$.
• Figure 5: The absolute value of the matrix elements of the projection matrix $P_{[2.1,3]}$ for $N=1000$, plotted on a (nonlinear) power scale. The left image shows the entire matrix $P_{[2.1,3]}$, while the right image is zoomed in to show the top left corner containing matrix entries $(x,y)$ with $x,y<100$. Most of the diagonal entries are generally close to $\frac{|I(N)|}{2\pi}=0.143\ldots$, and the large off-diagonal entries visually appear to follow the shape of the set $\widetilde{A}_{J,\delta,\gamma,N}^W$; outside this set the entries appear to be typically small, reflecting Proposition \ref{['prop:mp-sf']}.

Theorems & Definitions (41)

• Theorem 2.1: windowed spectral projection
• Remark 2.1
• Theorem 2.2: windowed generalized Weyl law
• Corollary 2.3: Weyl law/eigenvalue counting
• Remark 2.2
• Theorem 2.4: windowed quantum variance
• Corollary 2.5: windowed quantum ergodicity
• Remark 2.3
• Theorem 2.6: random quasimodes
• Remark 2.4