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

我们分析和实践了动态分布功能的研究系统包含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 functionalequations and their properties are investigated. 2)Microscopic mechanism of dissipation for aMultiBaker map with energy：We introduced a multibaker map with a coordinate corresponding to kinetic energy and studiednonequilibrium stationary states, relaxation modes, scaling-limit behaviors，time evolution towards the past.我们发现the dynamical reversibility is consistent withirreversible evolution of distribution functions. 3)Boundary元素方法（BEM） for 2d quantumbilliards and Fredholm theory：We studied the properties of a functional determinant D(E)BEM for 2d quantum billiards，and its semiclassical limit with the aid of the Fredholm theory. 4)Quantum nonequilibrium stationarystates:We studied the behaviors of a 1d noninteracting electron system placed between two perfectconductors with the aid of C d - 1-algebraic method. Two different stationary states are obtainedin the limit of t→±∞。By comparing their properties，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 Hamiltoniansuch as Gamow vectors. We investigated the properties of two interacting unstable states and必需mathematical backgrounds.Also we investigated other相关主题such as 6)Opticalproperties of carbon nanotubes7)Spectral properties of evolution operators of distribution functions for Hopf bifurcation andintermittent maps, 8)Transport properties of periodic intermittent maps，and 9)Stationary states for a reaction-diffusion type MultiBaker map。

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.