Around quantum ergodicity
英文摘要:
We discuss Shnirelman's Quantum Ergodicity Theorem, giving an outline of a proof and an overview of some of the recent developments in mathematical Quantum Chaos.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
Semyon Dyatlov
We discuss Shnirelman's Quantum Ergodicity Theorem, giving an outline of a proof and an overview of some of the recent developments in mathematical Quantum Chaos.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
继续探索
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 $|\log h|^{-α}$ for some $α>0$. Second, we show quantum ergodicity at polynomial scales $h^α$ for some $α>0$, in two special cases: $N\in S(\mathbb{N})$ of a full density subset $S(\mathbb{N})$ of integers and Hecke eigenbasis for all integers.
Quantum Ergodicity and Mixing
这是一篇为《数学物理百科全书》撰写的关于标题中主题的临时文章。
In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the torus ("cat maps'"). In Part II of the series, we construct quasimodes that are quantum ergodic but are not equidistributed at the logarithmical scales.
Quantum unique ergodicity for magnetic Laplacians on T^2
研究二维 torus 上磁 Laplacian 在高能情况下的量子唯一遍历性,即使经典流不满足遍历性条件。
We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on Riemannian surfaces with finitely many hyperbolic cusps and ergodic geodesic flow.
This paper is a continuation of Poiret-Robert-Thomann (2013) where we studied a randomisation method based on the Laplacian with harmonic potential. Here we extend our previous results to the case of any polynomial and confining potential $V$ on $\mathbb{R}^d$. We construct measures, under concentration type assumptions, on the support of which we prove optimal weighted Sobolev estimates on $\mathbb{R}^d$. This construction relies on accurate estimates on the spectral function in a non-compact configuration space. Then we prove random quantum ergodicity results without specific assumption on t…
This paper is a proceedings version of \cite{CHT-I}, in which we state a Quantum Ergodicity (QE) theorem on a 3D contact manifold, and in which we establish some properties of the Quantum Limits (QL). We consider a sub-Riemannian (sR) metric on a compact 3D manifold with an oriented contact distribution. There exists a privileged choice of the contact form, with an associated Reeb vector field and a canonical volume form that coincides with the Popp measure. We state a QE theorem for the eigenfunctions of any associated sR Laplacian, under the assumption that the Reeb flow is ergodic. The limi…
Upper bounds on the rate of quantum ergodicity
研究了具有遍历经典极限的量子系统本征函数的半经典行为。根据量子遍历性定理,几乎所有这些本征函数在弱意义上都是等分布的。当经典系统以一定速率遍历时,我们给出了量子遍历速率的$\abs{\ln\hbar}^{-1}$阶上界的一个简单推导。此外,当经典系统是弱混合时,我们得到了一个类似的跃迁振幅的界。这两个结果概括了以前的Zelditch。然后将结果推广到环面上的几类量子化映射,得到了扰动猫映射的对数速率和抛物映射的尖锐代数速率。
A new proof is given of Quantum Ergodicity for Eisenstein Series for cusped hyperbolic surfaces. This result is also extended to higher dimensional examples, with variable curvature.
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 of eigenfunctions. We also derive improvements on the size of the nodal sets. Our proof is based on a quantum ergodicity property of independent interest, which holds for families of symbols supported in balls whose radius shrinks at a logarithmic rate.
Quantum ergodicity for contact metric structures
本文在Reeb流遍历的假设下,利用半经典伪微分算子与Landau投影子,证明了接触度量流形上次拉普拉斯算子本征函数的量子遍历性定理。
Quantum ergodicity for restrictions to hypersurfaces
量子遍历定理指出,对于具有遍历经典流的量子系统,本征态平均均匀分布在能量面上。证明了若N是位置空间中满足简单动力学条件的超曲面,则本征态对N的约束也是量子遍历的。
Quantum Flux and Quantum Ergodicity for Cross Sections
本文定义量子通量范数,证明量子遍历本征函数在余维1子流形上的限制仍具量子遍历性,并与已知结果比较,进一步给出复化解析情形下的微局部类比。
The main theorem has been slightly generalized to include a larger class of symbols.
Quantum Ergodicity for a Class of Mixed Systems
我们研究高能量的Dirichlet拉普拉斯算子域台球流表现出混合动力学行为的特征函数。(More一般地,我们考虑半经典薛定谔算子与混合假设的哈密顿流。具体地说,我们假设台球流有一个不变的遍历组件,U,并研究缺陷措施,亩,正密度连续性的本征函数(更一般地说,几乎正交准模)。我们表明,任何缺陷措施相关联的这样一个子序列,当限制到U,满足μ = c μ_L,其中μ_L是刘维尔措施。这证明了珀西瓦尔猜想的一部分。
Semiclassical analysis, geometric representation and quantum ergodicity
本文结合半经典与几何量子化方法,证明了特定幺正平坦丛高频特征截面的均匀分布性质,推进了量子遍历性研究。
We are concerned with the geometric properties of the surface plasmon resonance (SPR). SPR is a non-radiative electromagnetic surface wave that propagates in a direction parallel to the negative permittivity/dielectric material interface. It is known that the SPR oscillation is very sensitive to the material interface. However, we show that the SPR oscillation asymptotically localizes at places with high magnitude of curvature in a certain sense. Our work leverages the Heisenberg picture of quantization and quantum ergodicity first derived by Shnirelman, Zelditch, Colin de Verdière and Helffer…
Quantum Ergodicity for Point Scatterers on Arithmetic Tori
我们证明了一个类似的Shnirelman,Zelditch和Colin de Verdiere的量子遍历定理的情况下,没有潜在的经典遍历。我们考虑的系统是Laplacian与三角洲潜力的平方环面。有两种类型的波函数:旧的本征函数的拉普拉斯算子,这是不受散射体,和新的本征函数具有对数奇异性的散射体的位置。证明了新特征函数的全密度子序列在相空间中均匀分布。我们的估计是统一的耦合参数,特别是等分布举行的弱和强耦合量子化的点散射。
In this paper we study the stochastic quantization problem on the two dimensional torus and establish ergodicity for the solutions. Furthermore, we prove a characterization of the $Φ^4_2$ quantum field on the torus in terms of its density under translation. We also deduce that the $Φ^4_2$ quantum field on the torus is an extreme point in the set of all $L$-symmetrizing measures, where $L$ is the corresponding generator.
We show that a Wigner induced random orthonormal basis of spherical harmonics is almost surely quantum ergodic. Here, a random basis is identified with an element of the product probability space of unitary groups, each endowed with the measure induced by the generalized Wigner ensemble. This yields a semi-classical realization of the probabilistic quantum unique ergodicity result for Wigner eigenvectors of Bourgade-Yau. At the same time, this generalizes a similar result due to Zelditch, who uses Haar measure on the unitary groups in defining the notion of a random basis.
Quantum ergodicity and semiclassical measures: mathematical results
本文回顾了混沌测地流下紧流形或欧几里得区域上拉普拉斯算子高频本征模的宏观分布,重点证明量子遍历性定理,并讨论量子唯一遍历性猜想及相关熵约束。
Quantum ergodicity for a point scatterer on the three-dimensional torus
考虑标准三维平坦环面上的点散射体(拉普拉斯算子受δ势扰动)。与拉普拉斯算子的本征函数在该点处消失,该算子有一组新的扰动本征函数。在最近的一篇论文中,作者能够证明所有的扰动本征函数均匀分布在位形空间。本文证明了几乎所有的本征函数在相空间中是均匀分布的,即证明了扰动本征函数的子空间的量子遍历性。Kurlberg和Ueberschär最近证明了二维环面上点散射体的类似结果。
Quantum ergodicity of boundary values of eigenfunctions: A control theory approach
考虑$M$,这是${\mathbb R}^d$中的一个有界区域,${\mathbb R}^d$是一个具有分段光滑边界的黎曼流形,并假设根据几何光学定律,与反射在边界上的测地流相关联的台球是遍历的。本文证明了Laplace算子在合理的边界条件下本征函数的边值在边界上渐近等分布,推广了Gérard,Leichtnam \cite{GeLe93-1}和Hassel,Zelditch \cite{HaZe02}在$M$凸性的附加假设下得到的结果.
The goal of this article is to draw new applications of small scale quantum ergodicity in nodal sets of eigenfunctions. We show that if quantum ergodicity holds on balls of shrinking radius $r(λ) \to 0$, then one can achieve improvements on the recent upper bounds of Logunov and Logunov-Malinnikova on the size of nodal sets, according to a certain power of $r(λ)$. We also show that the order of vanishing results of Donnelly-Fefferman and Dong can be improved. Since by the results of Han and Hezari-Rivière small scale QE holds on negatively curved manifolds at logarithmically shrinking rates, w…
In this thesis, we investigate quantum ergodicity for two classes of Hamiltonian systems satisfying intermediate dynamical hypotheses between the well understood extremes of ergodic flow and quantum completely integrable flow. These two classes are mixed Hamiltonian systems and KAM Hamiltonian systems. Hamiltonian systems with mixed phase space decompose into finitely many invariant subsets, only some of which are of ergodic character. It has been conjectured by Percival that the eigenfunctions of the quantisation of this system decompose into associated families of analogous character. The fi…
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