For manifolds with geodesic flow that is ergodic on the unit tangent bundle, the quantum ergodicity theorem implies that almost all Laplacian eigenfunctions become equidistributed as the eigenvalue goes to infinity. For a locally symmetric space with a universal cover that is a product of several upper half planes, the geodesic flow has constants of motion so it can not be ergodic. It is, however, ergodic when restricted to the submanifolds defined by these constants. In accordance, we show that almost all eigenfunctions become equidistributed on these submanifolds.
Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces
给出了Benjamini和Schramm意义下的固定谱窗和收敛到双曲平面的紧致双曲曲面序列的量子遍历性定理。这回答了Colin de Verdière提出的问题。我们的定理的灵感来自于Anantharaman和第一作者关于大型正则图的特征函数的结果。它特别适用于本征函数紧凑算术表面的水平方面，这将它连接到一个问题的纳尔逊马斯形式。证明是基于波传播的方法最近考虑布鲁克斯，林登施特劳斯和第一个命名的作者离散图。它不使用任何微局域分析，使它与通常的大本征值极限下的量子遍历性证明大不相同。此外，我们取代了波传播与重整化平均运营商在光盘上，这简化了分析，并允许我们利用一般的遍历定理的Nevo。作为这种方法的结果，我们几乎不需要观测值的正则性。
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.
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 paper we give an integral representation for the resolvent kernels with uniform magnetic field on the hyperbolic plane, as applications of our results we solve explicitly two times dependent Schrödinger equations with uniform magnetic field on the hyperbolic plane
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.
This paper deals with exact formulas for the resolvent kernels and exact solutions of time dependent Schrödinger equations under a uniform magnetic field on the hyperbolic half plane $\H$, and under a diatomic molecular Morse potential on the real line $\R$.
This paper deals with the $L_p$-spectrum of Schrödinger operators on the hyperbolic plane. We establish Lieb-Thirring type inequalities for discrete eigenvalues and study their dependence on $p$. Some bounds on individual eigenvalues are derived as well.
We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil-Petersson measure. The ensemble variance of the linear statistic was recently shown to coincide with that of the corresponding statistic in the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, in the double limit of first taking large genus and then shrinking size of the energy window. In this note we show that in this same limit, the energy variance for a typical surface …
