宇宙学弹球群与自伴BKL转移算符
Cosmological-Billiards Groups and self-adjoint BKL Transfer Operators
AI总结：
本研究针对4维时空下的两类宇宙学弹球，分类了其周期轨道与量子范畴内的本征值，得到半经典算符完整谱，发现宇宙学弹球的半经典转移算符为自伴算符，为Langlands Jaquet对应研究提供了新途径。
摘要：
根据BKL（Belinski Khalatnikov Lifshitz，别连斯基-哈拉特尼科夫-栗弗席兹）范式，当假设度规张量具有最普遍的对称性，且在趋近宇宙学奇点的渐近极限下将各点视为空间解耦时，宇宙学弹球作为爱因斯坦方程解的一种映射而出现。在4=3+1维时空下，研究考虑了两类宇宙学弹球：一类是描述纯引力的所谓“大弹球”，另一类是前者的对称性约化版本——“小弹球”，它是在考虑“对称墙”时得到的。通过这种方式，爱因斯坦场方程的解被映射为上庞加莱半平面（UPHP）中三角形弹球台边缘上弹球的（离散）庞加莱映射。弹球模群是分析常负曲率曲面上经典混沌系统动力学的理论框架。研究依据不同的对称商化机制，对两类弹球的周期轨道进行了分类，探究并概述了其与弹球模群所隐含描述的差异。在量子范畴内，对Maass波函数上BKL映射周期现象的本征值（即波函数根据量子BKL映射获得的符号）进行了分类，得到了作为周期轨道BKL映射的半经典算符的完整谱。与模群的情况不同，本研究表明宇宙学弹球的半经典转移算符不仅是作用于Maass波形的算符的伴随算符，而且二者是同一个自伴算符，由此提出了一种研究Langlands Jaquet对应关系的不同途径。
英文摘要：
Cosmological billiards arise as a map of the solution of the Einstein equations, when the most general symmetry for the metric tensor is hypothesized, and points are considered as spatially decoupled in the asymptotic limit towards the cosmological singularity, according to the BKL (Belinski Khalatnikov Lifshitz) paradigm. In $4=3+1$ dimensions, two kinds of cosmological billiards are considered: the so-called 'big billiard' which accounts for pure gravity, and the 'small billiard', which is a symmetry-reduced version of the previous one, and is obtained when the 'symmetry walls' are considered. The solution of Einstein field equations is this way mapped to the (discrete) Poincaré map of a billiard ball on the sides of a triangular billiard table, in the Upper Poincaré Half Plane (UPHP). The billiard modular group is the scheme within which the dynamics of classical chaotic systems on surfaces of constant negative curvature is analyzed. The periodic orbits of the two kinds of billiards are classified, according to the different symmetry-quotienting mechanisms. The differences with the description implied by the billiard modular group are investigated and outlined. In the quantum regime, the eigenvalues (i.e. the sign that wavefunctions acquire according to quantum BKL maps) for periodic phenomena of the BKL maps on the Maass wavefunctions are classified. The complete spectrum of the semiclassical operators which act as BKL map for periodic orbits is obtained. Differently form the case of the modular group, here it is shown that the semiclassical transfer operator for Cosmological Billiards is not only the adjoint operator of the one acting on the Maass waveforms, but that the two operators are the same \it{self-adjoint} operator, thus outlining a different approach to the Langlands Jaquet correspondence.
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