论文标题
理性台球中的波函数和能量光谱完全取决于其周期
Wave Functions and Energy Spectra in Rational Billiards Are Determined Completely by Their Periods
论文作者
论文摘要
理性的台球（RB）在经典上是可构成的，即它们在相空间中的轨迹位于多腹肌上。每个这样的多腹肌都可以展开为基本多边形模式（EPP）。然后，与每个RB相对应的有理台球表面（RBR）是通过定期分布EPP制成的无限镶嵌物。 RBR的周期与RB的定期轨道直接相关。结果表明，RB中Schrödinger方程（SE）的任何固定溶液（SS）都可以在整个RBR上扩展。然后，扩展的固定波函数（ESS）是在RBR及其周期的周期性的。相反，对于与EPP一致的每个边界条件系统（即Dirichlet或Neumann Ones或它们的混合物），可以找到所谓的固定前溶液（SP）（sps）的schrödinger方程，在RBR上定义并尊重其周期性结构以及其能量光谱。使用SPS可以轻松地通过SPS上的琐碎代数在其上的大多数边界条件下构造RB的SS。因此，它证明，由与每个RB相对应的SS的边界条件定义的能量光谱完全取决于$ 2G $ $独立的RBR时期是这些时期的均匀函数。由于所考虑的多边形台球的合理性，可以专门构建RBR。因此，本文中开发的方法可以看作是在RB中获取SS的新方法。可以针对一类RB明确构建SP，可以将EPP分解为一组平行的周期性轨道通道（POC）（POCDRB）。对于这样一类RB，各自的RBR可以作为具有周期性结构的标准多页面riemann表面构建。对于POCDRB，可以彻底完成对超车状态（SSS）存在的讨论。
The rational billiards (RB) are classically pseudointegrable, i.e. their trajectories in the phase space lie on multi-tori. Each such a multi-torus can be unfolded into elementary polygon pattern (EPP). A rational billiards Riemann surface (RBRS) corresponding to each RB is then an infinite mosaic made by a periodic distribution of EPP. Periods of RBRS are directly related to periodic orbits of RB. It is shown that any stationary solutions (SS) to the Schrödinger equation (SE) in RB can be extended on the whole RBRS. The extended stationary wave functions (ESS) are then periodic on RBRS with its periods. Conversely, for each system of boundary conditions (i.e. the Dirichlet or the the Neumann ones or their mixture) consistent with EPP one can find so called stationary pre-solutions (SPS) of the Schrödinger equation defined on RBRS and respecting its periodic structure together with their energy spectra. Using SPS one can easily construct SS of RB for most boundary conditions on it by a trivial algebra over SPS. It proves therefore that the energy spectra defined by the boundary conditions for SS corresponding to each RB are totally determined by $2g$ independent periods of RBRS being homogeneous functions of these periods. RBRS can be constructed exclusively due to the rationality of the polygon billiards considered. Therefore the approach developed in the present paper can be seen as a new way in obtaining SS to SE in RB. SPS can be constructed explicitly for a class of RB which EPP can be decomposed into a set of periodic orbit channel (POC) parallel to each other (POCDRB). For such a class of RB the respective RBRS can be built as a standard multi-sheeted Riemann surface with a periodic structure. For POCDRB a discussion of the existence of the superscar states (SSS) can be done thoroughly.
