Remarks on quantum ergodicity
Gabriel Riviere
证明了无边界光滑紧致黎曼流形上量子遍历定理的一个推广形式。我们应用它来证明一些渐近性质的典型特征函数的拉普拉斯算子的几何情况下，刘维测度是不是（或不知道是）遍历分布。
We prove a generalized version of the Quantum Ergodicity Theorem on smooth compact Riemannian manifolds without boundary. We apply it to prove some asymptotic properties on the distribution of typical eigenfunctions of the Laplacian in geometric situations where the Liouville measure is not (or not known to be) ergodic.
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We prove quantum ergodicity for certain orthonormal bases of $L^2(\mathbb{S}^2)$, consisting of joint eigenfunctions of the Laplacian on $\mathbb{S}^2$ and the discrete averaging operator over a finite set of rotations, generating a free group. If in addition the rotations are algebraic we give a quantified version of this result. The methods used also give a new, simplified proof of quantum ergodicity for large regular graphs.
Classical and quantum ergodicity on orbifolds
我们将Shnirelman、Colin de Verdière和Zelditch关于量子遍历性的经典结果推广到轨道折叠，证明了对于紧致轨道折叠X上的任何正的一阶自伴椭圆伪微分算子P，且具有正的主象征p，p的Hamilton流的遍历性蕴涵着算子P的量子遍历性。我们还证明了负的紧黎曼轨道上的测地流的遍历性。截面曲率
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.
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.
We outline some recent proofs of quantum ergodicity on large graphs and give new applications in the context of irregular graphs. We also discuss some remaining questions.
Localization in infinite billiards: a comparison between quantum and classical ergodicity
考虑在正$x$ -半轴、正$y$ -半轴和$f(x) = (x+1)^{-α}$、$α\in (1,2]$的图所包围的第一象限中的非紧台球。虽然Schnirelman定理成立，但位置$x$的量子平均在任何本征态上都是有限的，而经典遍历性要求$x$的经典时间平均是无界的。
Consider a sequence of finite regular graphs (GN) converging, in the sense of Benjamini-Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non-zero coupling constant $α$) and a symmetric potential U on the edges. We show that in the spectral regions where the infinite quantum tree has absolutely continuous spectrum, the eigenfunctions of the converging quantum graphs satisfy a quantum ergodicity theorem. In case $α$ = 0 and U = 0, the limit measure is the uniform measure on the edges. In general, it ha…
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.
We consider a sequence of finite quantum graphs with few loops, so that they converge, in the sense of Benjamini-Schramm, to a random infinite quantum tree. We assume these quantum trees are spectrally delocalized in some interval $I$, in the sense that their spectrum in $I$ is purely absolutely continuous and their Green's functions are well controlled near the real axis. We furthermore suppose that the underlying sequence of discrete graphs is expanding. We deduce a quantum ergodicity result, showing that the eigenfunctions with eigenvalues lying in $I$ are spatially delocalized.
Quantum Ergodicity on large hyperbolic surfaces for local and pseudolocal operators
本研究在超曲面上证明了量子混沌性，针对局部和伪局部算子，通过注入半径下界和谱间隙条件，证明量子方差消失，扩展了标量算子结果。
Quantum ergodicity on large regular graphs
我们提出了一个版本的量子遍历定理的大型正则图的固定价。这是“大多数”本征函数的离域性。我们考虑具有少量短圈的扩展图（例如随机大正则图）。我们的方法模仿流形上量子遍历性的证明：它使用正则树上的微局部分析。
Ergodicity in discrete-time quantum walks
本文研究了离散时间量子行走的遍历性，发现一维情况下单位算子的绝对连续谱与位置空间的动力学等分布完全等价，并在高维情况下提出'无重复图'谱性质作为全遍历和部分遍历的判据。
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 Dirichlet-truncated operators on $\mathbb{Z}^d$
本文证明了$\u210d^d$上邻接矩阵Dirichlet截断的量子遍历性，并推广到有限程观测和周期至多为二的周期Schrödinger算子，部分回答了McKenzie和Sabri提出的问题。
We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdière. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and the first-named author. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Linde…
We prove a result of delocalization for the Anderson model on the regular tree (Bethe lattice). When the disorder is weak, it is known that large parts of the spectrum are a.s. purely absolutely continuous, and that the dynamical transport is ballistic. In this work, we prove that in such AC regime, the eigenfunctions are also delocalized in space, in the sense that if we consider a sequence of regular graphs converging to the regular tree, then the eigenfunctions become asymptotically uniformly distributed. The precise result is a quantum ergodicity theorem.
We study the limiting behavior of Maass forms on Benjamini-Schramm convergent sequences of compact quotients of ${\rm SL}_d(\mathbb{R})/{\rm SO}(d)$, $d\ge 3$, whose spectral parameter stays in a fixed window. We prove a form of Quantum Ergodicity in this level aspect which extends results of Le Masson and Sahlsten to the higher rank case.
The main theorem has been slightly generalized to include a larger class of symbols.
We prove a quantum-ergodicity theorem on large graphs, for eigenfunctions of Schrödinger operators in a very general setting. We consider a sequence of finite graphs endowed with discrete Schrödinger operators, assumed to have a local weak limit. We assume that our graphs have few short loops, in other words that the limit model is a random rooted tree endowed with a random discrete Schrödinger operator. We show that absolutely continuous spectrum for the infinite model, reinforced by a good control of the moments of the Green function, imply "quantum ergodicity", a form of spatial delocalizat…
For periodic graph operators, we establish criteria to determine the overlaps of spectral band functions based on Bloch varieties. One criterion states that for a large family of periodic graph operators, the irreducibility of Bloch varieties implies no non-trivial periods for spectral band functions. This particularly shows that spectral band functions of discrete periodic Schrödinger operators on $\mathbb{Z}^d$ have no non-trivial periods, answering a question asked by Mckenzie and Sabri [Quantum ergodicity for periodic graphs. arXiv preprint arXiv:2208.12685 (2022)].
We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on Riemannian surfaces with finitely many hyperbolic cusps and ergodic geodesic flow.
We prove quantum ergodicity for a family of periodic Schrödinger operators $H$ on periodic graphs. This means that most eigenfunctions of $H$ on large finite periodic graphs are equidistributed in some sense, hence delocalized. Our results cover the adjacency matrix on $\mathbb{Z}^d$, the triangular lattice, the honeycomb lattice, Cartesian products and periodic Schrödinger operators on $\mathbb{Z}^d$. The theorem applies more generally to any periodic Schrödinger operator satisfying an assumption on the Floquet eigenvalues.
Quantum ergodicity for Eisenstein series on hyperbolic surfaces of large genus
本文给出了有限面积双曲面上量子平均绝对偏差的定量估计，证明了在 Benjamini-Schramm 收敛于双曲平面时 Laplacian 特征函数的去局部化，并指出该结果在 Mirzakhani 随机曲面模型中具有通用性。
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。
Quantum unique ergodicity for random bases of spectral projections
考虑Zelditch引入的随机波模型，研究了黎曼流形上典型准模的行为。利用指数矩方法，我们证明了在温和的增长假设下，随机波满足概率为1的量子唯一遍历性。
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.
Spectral convergence of hyperbolic surfaces and quantum ergodicity
该研究将Plancherel收敛推广到有限面积双曲曲面，建立其量子遍历性定理，涵盖退化情形，为退化双曲曲面的量子遍历性提供自然谱框架。
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…
Joint Quasimodes, Positive Entropy, and Quantum Unique Ergodicity
研究了紧致同余曲面上Laplacian算子与Hecke算子的联合拟模，给出了在相应的半经典测度的几乎每个遍历分量上保证正熵的拟模的阶的条件.与第二个命名的作者的测量分类结果一起，这意味着这些函数的量子唯一遍历性。我们的结果是最佳的空间的维度从准模式的构造。我们还研究了$\mathbb{H}\times\mathbb{H}$的紧不可约行列式上两个部分拉普拉斯算子的联合拟模序列的等分布。
The quantum unique ergodicity conjecture for thin sets
我们考虑一些类似的量子唯一遍历猜想测地线，horocycles，或“收缩”家庭的集合。特别是，我们证明了模拟的爱森斯坦级数的QUE猜想限制到无限测地线连接0和无限内的模块表面。
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。
We prove that the Hecke--Maass eigenforms for a compact arithmetic triangle group have a growing number of nodal domains as the eigenvalue tends to $+\infty$. More generally the same is proved for eigenfunctions on negatively curved surfaces that are even or odd with respect to a geodesic symmetry and for which Quantum Unique Ergodicity holds.
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 in the Benjamini--Schramm limit for locally symmetric spaces
研究在局部对称空间序列的Benjamini-Schramm极限下，不变微分算子的本征函数的平均去局域化特性。
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 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。
We investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture for half-integral weight automorphic forms. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for both half-integral weight holomorphic Hecke cusp forms for $Γ_0(4)$ lying in Kohnen's plus subspace and for half-integral weight Hecke Maaß cusp forms for $Γ_0(4)$ lying in Kohnen's plus subspace. By combining the former result along with an argument of Rudnick, it follows that under GRH the zeros of these holomorphic Hecke cusp equidistribute with respect to hyperbolic measure on $Γ_0(4)\backslash \math…
A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity
本研究针对谱截断谱三元组提出非交换积分的简单近似，推导得到非交换几何的Szegő极限公式，建立其与态密度的关联，提出紧谱三元组测地流遍历性的定义，揭示量子遍历性与Connes积分公式的紧密联系。
Quantum unique ergodicity for parabolic maps
研究了环面的量子化遍历映射的遍历性质。已知这些满足量子遍历性：对于几乎所有的本征态，量子可观测量的期望值收敛到经典相空间平均值，相对于相应的经典可观测量的刘维尔测度。本征态可能存在的任何异常双折射是一个重要的问题，到目前为止，在任何例子中都没有解决。不存在例外的连续性被称为量子唯一遍历性（quantum unique ergodicity，QUE）。我们提出的第一个例子的地图，满足QUE：无理斜平移的两个环面，抛物类似物阿诺德的猫地图。这些映射是经典的唯一遍历的，不是混合的。关键的一步是找到一个符合量子-经典对应原理的量子化配方。除了证明这些地图的QUE，我们也给结果的收敛速度相空间平均。我们给出的上限，我们表明是最优的。我们构造这些地图的收敛速度是任意慢的特殊的例子。
The problem of quantum unique ergodicity (QUE) of weight 1/2 Eisenstein series for Γ_0(4) leads to the study of certain double Dirichlet series involving GL2 automorphic forms and Dirichlet characters. We study the analytic properties of this family of double Dirichlet series (analytic continuation, convexity estimate) and prove that a subconvex estimate implies the QUE result.
Patterson-Sullivan distributions and quantum ergodicity
本文讨论了紧致双曲曲面$X_Γ$上拉普拉斯算子的本征函数$ϕ_{ir_j}$的两种相空间分布：（1）Wigner分布$\int_{S^*\X} a dW_{ir_j}=< Op(a)ϕ_{ir_j}, ϕ_{ir_j}>_{L^2(\X)}$，它产生于量子混沌。它们在波群下是不变的。(2)Patterson-Sullivan分布$PS_{ir_j}$，它是动态zeta函数$\lcal(s; a): = \sum_γ\frac{e^{-sL_γ}}{1-e^{-L_γ}} \int_{γ_0} a$（其中和在闭测地线上运行）在极点$s = {1/2} + ir_j$的残差。它们在测地流下是不变的。我们证明，这些分布（适当归一化时）是渐近相等的$r_j \to \infty$。我们也给出了它们之间的确切关系。这种对应给出了双曲曲面上经典动力学和量子动力学之间的一种新的关系，从而给出了经典遍历论中量子遍历性的一种表述。
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.
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…
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.
We study the ergodic properties of Schrödinger operators on a compact connected Riemannian manifold $M$ without boundary in case that the underlying Hamiltonian system possesses certain symmetries. More precisely, let $M$ carry an isometric and effective action of a compact connected Lie group $G$. Relying on an equivariant semiclassical Weyl law proved in Part I of this work, we deduce an equivariant quantum ergodicity theorem under the assumption that the symmetry-reduced Hamiltonian flow on the principal stratum of the singular symplectic reduction of $M$ is ergodic. In particular, we obtai…
Quantum Unique Ergodicity for Cayley graphs of quasirandom groups
该研究针对拟随机群的Cayley图，证明高度拟随机群的任意Cayley图存在邻接算子的标准正交特征基，其特征函数的量子概率测度在群的非小子集上的质量占比接近理论正确比例，推进了量子唯一遍历性在群图中的研究。
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