Corpus ID: 119739301

Small scale quantum ergodicity in cat maps. I

@article{Han2018SmallSQ,
  title={Small scale quantum ergodicity in cat maps. I},
  author={X. Han},
  journal={arXiv: Mathematical Physics},
  year={2018}
}
  • X. Han
  • Published 29 October 2018
  • Mathematics, Physics
  • arXiv: Mathematical Physics
In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the torus ("cat maps"). In Part I of the series, we prove quantum ergodicity at various scales. Let $N=1/h$, in which $h$ is the Planck constant. First, for all integers $N\in\mathbb{N}$, we show quantum ergodicity at logarithmical scales $|\log h|^{-\alpha}$ for some $\alpha>0$. Second, we show quantum ergodicity at polynomial scales $h^\alpha$ for some $\alpha>0$, in two special cases: $N\in S… Expand

References

SHOWING 1-10 OF 48 REFERENCES
On the Entropy of Quantum Limits for 2-Dimensional Cat Maps
We study semiclassical measures, or quantum limits, for quantized hyperbolic automorphisms of $${\mathbb{T}^2}$$ . We show that any quantum limit has the following property: if a weight α is carriedExpand
On Quantum Ergodicity for Linear Maps of the Torus
Abstract: We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (“cat maps”). We show that there is a density one sequence of integers so that as N tends to infinityExpand
Scarring of quasimodes on hyperbolic manifolds
Let $N$ be a compact hyperbolic manifold, $M\subset N$ an embedded totally geodesic submanifold, and let $-\hbar^2\Delta_{N}$ be the semiclassical Laplace--Beltrami operator. For anyExpand
On the Rate of Quantum Ergodicity for Quantised Maps
Abstract.We study the distribution of expectation values and transition amplitudes for quantised maps on the torus. If the classical map is ergodic then the variance of the distribution ofExpand
Scarred Eigenstates for Quantum Cat Maps of Minimal Periods
In this paper we construct a sequence of eigenfunctions of the ``quantum Arnold's cat map'' that, in the semiclassical limit, shows a strong scarring phenomenon on the periodic orbits of theExpand
Ergodic billiards that are not quantum unique ergodic
Partially rectangular domains are compact two-dimensional Riemannian manifolds $X$, either closed or with boundary, that contain a flat rectangle or cylinder. In this paper we are interested inExpand
Strong Scarring of Logarithmic Quasimodes
We consider a semiclassical (pseudo)differential operator on a compact surface $(M,g)$, such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit $\gamma$ atExpand
Small Scale Equidistribution of Random Eigenbases
  • X. Han
  • Mathematics, Physics
  • 2015
We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e., eigenbases) on a compact manifold $${{\mathbb M}}$$M. Assume that the group of isometries actsExpand
Joint quasimodes, positive entropy, and quantum unique ergodicity
We study joint quasimodes of the Laplacian and one Hecke operator on compact congruence surfaces, and give conditions on the orders of the quasimodes that guarantee positive entropy on almost everyExpand
$L^p$ norms, nodal sets, and quantum ergodicity
For small range of $p>2$, we improve the $L^p$ bounds of eigenfunctions of the Laplacian on negatively curved manifolds. Our improvement is by a power of logarithm for a full density sequence ofExpand
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