量子台球系统的一种有效有限元方法

边界积分方法在混沌量子台球中的应用

Boundary integral method applied in chaotic quantum billiards

边界积分法（BIM）是亥姆霍兹方程的一种形式，适用于数值离散化以求解量子台球。本文是一个广泛的数值调查BIM在各种量子台球，可积（圆形，矩形），KAM系统（Robnik台球）和完全混乱（遍历，如体育场，西奈台球和心形台球）。在理论方面，我们指出了一些严重的缺陷，在推导BIM的文献中，并显示如何最终公式（但这是正确的）应得出一个健全的方式，我们还认为，一个简单的头脑应用BIM在非凸几何提出了严重的困难，甚至失败。在数值方面，我们分析了本征能的系统误差$ΔE$的平均绝对值（以平均能级间距为单位）与离散密度的比例关系（$b$ =一个德布罗意波长内边界上的数值节点数），并且我们发现在所有情况下误差服从幂律$ <|ΔE|> = A b^{-α}$，其中$ α$（以及$A$）因情况而异（它不是通用的），并且受到非凸几何中外弦存在的强烈影响，而经典混沌的程度似乎实际上无关紧要。我们对BIM的半经典极限进行了评述，并对非凸几何中正确的半经典极限公式提出了建议。

一种寻找混沌系统轨道的替代方法

Alternative method to find orbits in chaotic systems

本文提出了一种新的方法，它应用有序符号动力学来寻找混沌系统中的不稳定周期和非周期轨道。该方法简单有效，并已成功地应用于Hénon映射、圆盘台球、体育场台球、楔形台球、抗磁开普勒问题、共线氦原子和吸引势系统等不同的系统.该方法似乎比以前使用的方法更好。

台球中量子运动的形态学图像分析

Morphological Image Analysis of Quantum Motion in Billiards

形态学图像分析应用于量子粒子在二维和三维台球运动中的概率分布的时间演化。它表明，时间平均欧拉特性的概率密度提供了一个定义良好的量来区分经典的可积和不可积台球。在三维中，概率密度的时间平均平均宽度也可用于此目的。

Interaction Matrix Element Fluctuations in Ballistic Quantum Dots: Dynamical Effects

We study matrix element fluctuations of the two-body screened Coulomb interaction and of the one-body surface charge potential in ballistic quantum dots, comparing behavior in actual chaotic billiards with analytic results previously obtained in a normalized random wave model. We find that the matrix element variances in actual chaotic billiards typically exceed by a factor of 3 or 4 the predictions of the random wave model, for dot sizes commonly used in experiments. We discuss dynamical effects that are responsible for this enhancement. These dynamical effects have an even more striking effe…

任意形状台球的本征态

Eigenstates for billiards of arbitrary shapes

提出了一种确定n维台球本征态的新算法。它是基于应用柯西定理确定的零空间的边界重叠矩阵。该方法是免费的限制与形状的台球，甚至可以适用于非凸几何形状，其他算法面临的困难。此外，它不受存在的特征值退化，这是另一个严重的缺点，许多方法。在本文中，我们应用该算法的几个简单的情况下，存在的解析解。数值解已被调查的情况下，环形台球。

混沌台球半经典量子化中的高阶hbar修正

Higher-order hbar corrections in the semiclassical quantization of chaotic billiards

在物理系统的周期轨道量子化中，通常只考虑首阶hbar对态密度的贡献。因此，通过构造，由半经典迹公式导出的本征值一般只在hbar的最低阶才与精确量子本征值一致。在不同的理论工作中，迹公式被推广到更高阶的hbar。然而，问题仍然存在，如何从扩展迹公式中实际计算本征值，因为即使包括hbar校正，周期轨道和仍然不收敛于物理域。对于最低阶的半经典迹公式的收敛问题可以优雅地，普遍地，规避应用技术的谐波反演。在本文中，我们将展示如何，对于一般的标度混沌系统，也高阶hbar校正Gutzwiller公式可以包括在谐波反演计划，并证明，校正的半经典特征值可以计算，尽管收敛问题。该方法被应用到开放的三盘散射系统，作为一个混沌系统的原型。

Stable and unstable regimes in higher-dimensional convex billiards with cylindrical shape

We introduce a class of convex, higher-dimensional billiard models which generalise stadium billiards. These models correspond to the free motion of a point-particle in a region bounded by cylinders cut by planes. They are motivated by models of particles interacting via a string-type mechanism, and confined by hard walls. The combination of these elements may give rise to a defocusing mechanism, similar to that in two dimensions, which allows large chaotic regions in phase space. The remaining part of phase space is associated with marginally stable behaviour. In fact periodic orbits in these…

Spin Chaos Manifestation in a Driven Quantum Billiard with Spin-Orbit Coupling

The coupling of orbital and spin degrees of freedom is the source of many interesting phenomena. Here, we study the electron dynamics in a quantum billiard --a mesoscopic rectangular quantum dot-- with spin-orbit coupling driven by a periodic electric field. We find that both the spatial and temporal profiles of the observables demonstrate the transition to chaotic dynamics with qualitative modifications of the power spectra and patterns of probability and spin density. The time dependence of the wavefunctions and spin density distributions indicates spin-charge separation {seen in the decay o…

具有点状散射体的三维量子台球的光谱特性

Spectral Properties of Three-Dimensional Quantum Billiards with a Pointlike Scatterer

我们研究了三维量子台球的光谱特性，其中有一个单一的点状散射体。研究发现，当形式强度的倒数$\bar{v}^{-1}$位于一个宽度随能量呈抛物线增长的带内时，谱表现出混沌（类随机矩阵）特性。这意味着在非常高的能量下，谱变得像随机矩阵，而与形式强度的值无关。数值实验与矩形框的预测证实。点状散射体的研究结果适用于一个小的，但有限大小的杂质的情况。我们澄清了适当的程序，其零尺寸的限制，其中涉及非平凡的分歧。从这个角度也回顾了以前已知的结果，在一维和二维量子台球内的小杂质。

量子混沌的实际和算法表现

Practical and algorithmical manifestations of quantum chaos

量子混沌还表现在求解薛定谔方程的方法的算法复杂性上，包括数值方法。在这方面的贡献，我们解决的问题，计算本征能和本征态的各种数值方法适用于2维通用台球系统。特别是，我们分析的精度（误差）的离散化的给定的数值方法的密度的依赖。我们这样做几个不同的台球形状，特别是Robnik台球。我们研究了边界积分法和平面波分解法的数值误差作为离散化参数$b$的函数，根据定义，$b$是边界上每一个德布罗意波长（弧长）间隔的离散节点的数量。对于边界积分方法，我们发现，在每个$λ$的误差尺度为幂律$<|ΔE|> = A b^{-α}$，其中$α$是$λ$的强函数：在KAM类制度$0\le λ\le 1/4$，它是大的，接近3.5，但接近$λ= 1/4$，它几乎不连续地变化，几乎没有任何大于零。这是因为台球从$λ=1/4$开始变为非凸的。对于平面波分解方法，我们发现在固定的$b$下，误差$<|ΔE|>$的平均绝对值与经典混沌密切相关，而误差$<|ΔE|>$随经典混沌的增加而急剧增加.该属性也适用于布尼莫维奇体育场和西奈台球场。

Distribution of Scattering Matrix Elements in Quantum Chaotic Scattering

Scattering is an important phenomenon which is observed in systems ranging from the micro- to macroscale. In the context of nuclear reaction theory the Heidelberg approach was proposed and later demonstrated to be applicable to many chaotic scattering systems. To model the universal properties, stochasticity is introduced to the scattering matrix on the level of the Hamiltonian by using random matrices. A long-standing problem was the computation of the distribution of the off-diagonal scattering-matrix elements. We report here an exact solution to this problem and present analytical results f…

Superstatistics in Random Matrix Theory

Random matrix theory (RMT) provides a successful model for quantum systems, whose classical counterpart has a chaotic dynamics. It is based on two assumptions: (1) matrix-element independence, and (2) base invariance. Last decade witnessed several attempts to extend RMT to describe quantum systems with mixed regular-chaotic dynamics. Most of the proposed generalizations keep the first assumption and violate the second. Recently, several authors presented other versions of the theory that keep base invariance on the expense of allowing correlations between matrix elements. This is achieved by s…

Analytical Approach to Noise Effects on Synchronization in a System of Coupled Excitable Elements

We report relationships between the effects of noise and applied constant currents on the behavior of a system of excitable elements. The analytical approach based on the nonlinear Fokker-Planck equation of a mean-field model allows us to study the effects of noise without approximations only by dealing with deterministic nonlinear dynamics . We find the similarity, with respect to the occurrence of oscillations involving subcritical Hopf bifurcations, between the systems of an excitable element with applied constant currents and mean-field coupled excitable elements with noise.