Quantum unique ergodicity for SL_2(Z)\H
K. Soundararajan
我们消除了模群的大本征值的Hecke-Maass形式的“质量逃逸”的可能性。结合Lindenstrauss的工作，建立了Rudnick和Sarnak关于模曲面SL_2（Z）\H的量子唯一遍历性猜想.
We eliminate the possibility of "escape of mass" for Hecke-Maass forms of large eigenvalue for the modular group. Combined with the work of Lindenstrauss, this establishes the Quantum Unique Ergodicity conjecture of Rudnick and Sarnak for the modular surface SL_2(Z)\H.
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In this paper we give a new proof of the Quantum Unique Ergodicity conjecture for holomorphic integral weight modular forms on the upper half plane. The proof requires only partial results towards the Ramanujan conjecture and the shifted convolution problem. Furthermore the proof is applicable to a wider class of cusp forms other than Hecke eigenforms. We also prove some interesting corollaries, particularly towards the Lehmer's conjecture on the non vanishing of the Fourier coefficients.
Arithmetic Quantum Unique Ergodicity for Symplectic Linear Maps of the Multidimensional Torus
我们期待的期望值的量化线性辛映射的多维环面和它们的分布在半经典极限。我们构建的超级疤痕，是稳定的算术对称的系统和本地化周围不变的流形。我们证明了这些超疤痕存在时，只有有各向同性的有理子空间，线性映射下不变。在没有疤痕的情况下，我们计算了去对称化系统的矩阵元的涨落方差，并对它们的极限分布提出了一个猜想。
We prove quantum unique ergodicity for a subspace of the continuous spectrum spanned by the degenerate Eisenstein Series on GL(n).
We prove a variety of quantum unique ergodicity results for Eisenstein series in the level aspect. A new feature of this variant of QUE is that the main term involves the logarithmic derivative of a Dirichlet $L$-function on the $1$-line. A zero of this $L$-function near the $1$-line can thus have a distorting effect on the main term. We obtain quantitative control on the test function and thereby prove an asymptotic formula in the level aspect version of the problem with test functions of shrinking support. Surprisingly, this asymptotic formula shows some obstruction to equidistribution that …
We consider some analogs of the quantum unique ergodicity conjecture for geodesics, horocycles, or ``shrinking'' families of sets. In particular, we prove the analog of the QUE conjecture for Eisenstein series restricted to the infinite geodesic connecting 0 and infinity inside the modular surface.
Quantum Unique Ergodicity for Eisenstein Series on Bruhat-Tits Buildings
本文在水平方向上证明了函数域上Eisenstein级数的量子唯一遍历性猜想，通过改进Luo-Sarnak机制并运用谱分解分别处理尖点与Eisenstein贡献。
We show that Landau-Siegel zeros for Dirichlet $L$-functions do not exist or quantum unique ergodicity for $\mathrm{GL}_2$ Hecke-Maass newforms holds with an effective rate of convergence. This follows from a more general result: Landau-Siegel zeros of Dirichlet $L$-functions repel the zeros of all other automorphic $L$-functions from the line $\mathrm{Re}(s)=1$.
Arithmetic unique ergodicity for flat vector bundles
本文证明了算术曲面上特定酉平坦丛的泡利-薛定谔自旋算子的高频特征截面的量子唯一遍历性的均匀版本。
Quantum Unique Ergodicity for Eisenstein Series on the Hilbert Modular Group over a Totally Real Field
W. Luo和P.Sarnak证明了$\rm{PSL}(2,\mathbb{Z}) \backslash H$上Eisenstein级数的量子唯一遍历性。本文将他们的结果推广到$\rm{PSL}(2,O) \backslash H^n$上的Eisenstein级数，其中$O$是$Q$上的$n$度全实域上的整数环，窄类数为1。埃弗拉特本文还对非全纯Hilbert模形式上的Hecke算子理论作了简要的讨论。
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.
Arithmetic quantum unique ergodicity for products of hyperbolic $2$- and $3$-spaces
本文证明了双曲2-与3-空间乘积商空间上Hecke-Maass形式的算术量子唯一遍历性猜想，通过轨道维数归纳法排除了极限测度集中于真子群闭轨道的可能性。
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。
We prove the arithmetic quantum unique ergodicity (AQUE) conjecture for non-degenerate sequences of Hecke eigenfunctions on quotients $Γ\backslash G/K$, where $G\simeq\mathrm{PGL}_{d}(\mathbb{R})$, $K$ is a maximal compact subgroup of $G$ and $Γ<G$ is a lattice associated to a division algebra over $\mathbb{Q}$ of prime degree $d$. More generally, we introduce a new method of proving positive entropy of quantum limits, which applies to higher-rank groups. The result on AQUE is obtained by combining this with a measure-rigidity theorem due to Einsiedler-Katok, following a strategy first pioneer…
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.
On Quantum unique ergodicity for locally symmetric spaces I
我们构造了局部对称空间的等变微局部提升。换句话说，我们演示了如何解除，在一个“半正则”的方式，局部对称空间上的Cartan-不变测度的本征函数措施的限制在一个适当的束。该构造使用了半单实李群的表示论的基本特征，并且可以被认为是Zelditch结果从上半平面到所有非紧型局部对称空间的推广。这将在续集中应用于解决某些局部对称空间上的量子唯一遍历问题。
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…
Subconvexity Implies Effective Quantum Unique Ergodicity for Hecke-Maaß Cusp Forms on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathrm{SL}_2(\mathbb{R})$
本文证明 Hecke-Maaß 尖点形式的相空间量子唯一遍历性可由三重积 $L$-函数的次凸界推出，填补了相空间层面此前仅有 Eisenstein 级数结果的空白，证明基于 Watson-Ichino 公式及 Whittaker 函数积分的精细分析。
On quantum ergodicity for linear maps of the torus
我们证明了环面上线性双曲映射（“猫映射”）的量子遍历性的一个强版本。我们证明了存在一个密度为1的整数序列，使得当N沿该序列趋于无穷大时，在逆普朗克常数N处的量子传播子的所有本征函数都是均匀分布的。论证中的一个关键步骤是证明，对于模群中的双曲矩阵，存在一个密度为1的整数序列N，其模N的阶（或周期）略大于N的平方根。
We prove an analogue of Shnirelman, Zelditch and Colin de Verdiere's Quantum Ergodicity Theorems in a case where there is no underlying classical ergodicity. The system we consider is the Laplacian with a delta potential on the square torus. There are two types of wave functions: old eigenfunctions of the Laplacian, which are not affected by the scatterer, and new eigenfunctions which have a logarithmic singularity at the position of the scatterer. We prove that a full density subsequence of the new eigenfunctions equidistribute in phase space. Our estimates are uniform with respect to the cou…
We study a refinement of the quantum unique ergodicity conjecture for shrinking balls on arithmetic hyperbolic manifolds, with a focus on dimensions $ 2 $ and $ 3 $. For the Eisenstein series for the modular surface $\mathrm{PSL}_2( {\mathbb Z}) \backslash \mathbb{H}^2$ we prove failure of quantum unique ergodicity close to the Planck-scale and an improved bound for its quantum variance. For arithmetic $ 3 $-manifolds we show that quantum unique ergodicity of Hecke-Maaß forms fails on shrinking balls centered on an arithmetic point and radius $ R \asymp t_j^{-δ} $ with $ δ> 3/4 $. For $ \mathr…
We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of $GL(2,\mathbb{Q}_p)$ for automorphic forms belonging to the principal series. We interpret this conclusion in terms of the equidistribution of eigenfunctions on covers of a fixed regular graph or along nested sequences of regular graphs. Our results are the first of their kind on any p-adic arithmetic quotient. They may be understood as analogues of Lindenstrauss's theorem on the equidistribution of Maass forms on a compact arithmetic surface. The new ingredients here include the introduction of a repre…
Quantum ergodicity of Eisenstein series for Bianchi groups
本文证明了Bianchi群上算术超曲面3流形上Eisenstein系列的量子混浊性，扩展了Koyama的工作，并首次在数域上非平凡类群的情况下建立了Eisenstein系列的量子混浊性。
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