Semiclassical analysis, geometric representation and quantum ergodicity
Minghui Ma, Qiaochu Ma
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。
本文利用半经典与几何量子化的混合方法，证明了特定系列幺正平坦丛的高频特征截面的均匀分布性质。
In this paper, we prove the equidistribution property of high-frequency eigensections of a certain series of unitary flat bundles, using the mixture of semiclassical and geometric quantizations.
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Within the framework of Weyl calculus we establish a quantum-classical correspondence for the time evolution of observables generated by a Dirac-Hamiltonian. This includes a semiclassical separation of particles and antiparticles. We then prove quantum ergodicity for Dirac-Hamiltonians under the condition that a skew product of the classical relativistic translational motion and relativistic spin precession is ergodic.
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。
This undergraduate thesis is concerned with developing the tools of differential geometry and semiclassical analysis needed to understand the the quantum ergodicity theorem of Schnirelman (1974), Zelditch (1987), and Colin de Verdière (1985) and the quantum unique ergodicity conjecture of Rudnick and Sarnak (1994). The former states that, on any Riemannian manifold with negative curvature or ergodic geodesic flow, the eigenfunctions of the Laplace-Beltrami operator equidistribute in phase space with density 1. Under the same assumptions, the latter states that the eigenfunctions induce a seque…
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We study the semiclassical behaviour of eigenfunctions of quantum systems with ergodic classical limit. By the quantum ergodicity theorem almost all of these eigenfunctions become equidistributed in a weak sense. We give a simple derivation of an upper bound of order $\abs{\ln\hbar}^{-1}$ on the rate of quantum ergodicity if the classical system is ergodic with a certain rate. In addition we obtain a similar bound on transition amplitudes if the classical system is weak mixing. Both results generalise previous ones by Zelditch. We then extend the results to some classes of quantised maps on th…
Quantum ergodicity, which expresses the semiclassical convergence of almost all expectation values of observables in eigenstates of the quantum Hamiltonian to the corresponding classical microcanonical average, is proven for non-relativistic quantum particles with spin 1/2. It is shown that quantum ergodicity holds, if a suitable combination of the classical translational dynamics and the spin dynamics along the trajectories of the translational motion is ergodic.
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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。
L2 geometric ergodicity for the kinetic Langevin process with non-equilibrium steady states
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We look at the expectation values for quantized linear symplectic maps on the multidimensional torus and their distribution in the semiclassical limit. We construct super-scars that are stable under the arithmetic symmetries of the system and localize around invariant manifolds. We show that these super-scars exist only when there are isotropic rational subspaces, invariant under the linear map. In the case where there are no such scars, we compute the variance of the fluctuations of the matrix elements for the desymmetrized system, and present a conjecture for their limiting distributions.
We study the ergodic properties of eigenfunctions of Schrödinger operators on a closed connected Riemannian manifold $M$ in case that the underlying Hamiltonian system possesses certain symmetries. More precisely, let $M$ carry an isometric effective action of a compact connected Lie group $G$. We prove an equivariant quantum ergodicity theorem assuming that the symmetry-reduced Hamiltonian flow on the principal stratum of the singular symplectic reduction of $M$ is ergodic. We deduce the theorem by proving an equivariant version of the semiclassical Weyl law, relying on recent results on sing…
In this paper we establish the ergodicity of Langevin dynamics for simple two-particle system involving a Lennard-Jones type potential. To the best of our knowledge, this is the first such result for a system operating under this type of potential. Moreover we show that the dynamics are {\it geometrically} ergodic (have a spectral gap) and converge at a geometric rate. Methods from stochastic averaging are used to establish the existence of a Lyapunov function. The existence of a Lyapunov function in this setting seems resistant to more traditional approaches. This is a corrected version of th…
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 consider a simple model for the fluctuating hydrodynamics of a flexible polymer in dilute solution, demonstrating geometric ergodicity for a pair of particles that interact with each other through a nonlinear spring potential while being advected by a stochastic Stokes fluid velocity field. This is a generalization of previous models which have used linear spring forces as well as white-in-time fluid velocity fields. We follow previous work combining control theoretic arguments, Lyapunov functions, and hypo-elliptic diffusion theory to prove exponential convergence via a Harris chain argume…
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Infinite ergodicity for geometric Brownian motion
本文研究几何布朗运动及其推广的渐近分布特性，通过无限遍历性方法探讨其在不同离散化参数下的概率分布行为。
The Quantum Unique Ergodicity (QUE) conjecture of Rudnick-Sarnak is that every eigenfunction phi_n of the Laplacian on a manifold with uniformly-hyperbolic geodesic flow becomes equidistributed in the semiclassical limit (eigenvalue E_n -> infinity), that is, `strong scars' are absent. We study numerically the rate of equidistribution for a uniformly-hyperbolic Sinai-type planar Euclidean billiard with Dirichlet boundary condition (the `drum problem') at unprecedented high E and statistical accuracy, via the matrix elements <phi_n, A phi_m> of a piecewise-constant test function A. By collectin…
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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。
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Finite-gap potentials as a semiclassical limit of the thermodynamic Bethe Ansatz
本文展示热力学Bethe Ansatz方程的半经典极限自然重构了有限-gap周期势的代数几何谱，通过行波解说明，量子场论的Bethe根分布对应于椭圆黎曼面的二次微分结构，谱的解析结构由Dynkin图及其大秩极限决定。
Hodge Topology of Semiclassical Transport: A Coordinate-Free Geometric Framework for the Anomalous Hall Effect and Non-Linear Berry Dipole
利用Brillouin区的Hodge-de Rham分解建立拓扑带中反常输运的坐标无关微分几何框架，通过正则化Berry曲率2-形式得到全局光滑的代理势，统一线性与非线性输运的几何描述，并证明Hodge势与最大局域Wannier函数规范的联系。
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