Analytical Research on the Eigenvalue Problem of the Evolution Operator of Distribution Functions for Nonlinear Systems exhibiting Chaos and Bifurcation

我们研究了动态系统中包括混沌系统的分布函数的行为，分析和实际上是。获得的结果如下： 1）SRB对不同贝克地图的测量：对de Rham的功能方程及其属性的援助进行了分析。2）具有能量的多贝克分布图的微观机制：我们引入了一个多贝克分布图，与动力学能量和研究的非稳定性静态状态、放松模式、缩放限制行为、时间演变到过去。我们发现动态可逆性与分布函数的不可逆进化是一致的。3）二维量子亿和弗雷德里克理论的边界元素方法（BEM）：我们研究了D（E）的功能决定性属性，BEM在二维量子亿中出现的地方，以及对弗雷德里克理论的辅助的半严格限制。4）Quantum nonekilibrium stationary states ： We studied the beManagement ors of a 1 d noninteracting electron system placed between two perfect conductors with the aid of CイD1* イエD1-algebraic method。Two different stationary states are obtained in the limit of t。通过比较他们的属性，动态的可逆性似乎与国家的不可逆演变是一致的。（5）不稳定的量子状态：不稳定的量子状态可以在Gamow矢量的术语中表示出来。我们研究了两个相互作用的不稳定状态和必要的物理背景的特性，我们还研究了其他相关主题，如6）碳纳米管的光学特性， 7）霍普双聚和间断图的分布函数的光谱特性， 8）周期间断图的传输特性，以及9）反应差异型多贝克图的稳定状态。

We studied analytically and exactly the behaviors of the distribution functions for dynamical systems including chaotic systems. Following results are obtained : 1)SRB measures for dissipative Baker maps : Invariant distributions and physical measures are constructed with the aid of de Rham's functional equations and their properties are investigated. 2)Microscopic mechanism of dissipation for a MultiBaker map with energy : We introduced a multibaker map with a coordinate corresponding to kinetic energy and studied nonequilibrium stationary states, relaxation modes, scaling-limit behaviors, time evolution towards the past. We found that the dynamical reversibility is consistent with irreversible evolution of distribution functions. 3)Boundary element methods (BEM) for 2d quantum billiards and Fredholm theory : We studied the properties of a functional determinant D(E), which appears in BEM for 2d quantum billiards, and its semiclassical limit with the aid of the Fredholm theory. 4)Quantum nonequilibrium stationary states : We studied the behaviors of a 1d noninteracting electron system placed between two perfect conductors with the aid of CィイD1*ィエD1-algebraic method. Two different stationary states are obtained in the limit of t →±∞. By comparing their properties, the dynamical reversibility is shown to be consistent with irreversible evolution of states. 5)Unstable quantum states : Unstable quantum states can be represented in terms of generalized eigenfunctions of the Hamiltonian such as Gamow vectors. We investigated the properties of two interacting unstable states and necessary mathematical backgrounds.Also we investigated other related topics such as 6)Optical properties of carbon nanotubes, 7)Spectral properties of evolution operators of distribution functions for Hopf bifurcation and intermittent maps, 8)Transport properties of periodic intermittent maps, and 9)Stationary states for a reaction-diffusion type MultiBaker map.