标题： 动力系统和量子力学中的相位锁定 显示英文标题

标题： Phase-locking in dynamical systems and quantum mechanics

摘要： 在本研究中，我们讨论将环面上的动力系统与Hill方程联系起来的普吕弗变换，该方程被解释为参量振荡器的运动方程或具有周期势的薛定谔方程。 环面上动力系统的相位锁定域结构映射到Hill方程的带隙结构。 对于参量振荡器，我们提供了非绝热汉纳角与相应动力系统的庞加莱旋转数之间的关系。 从量子力学的角度来看，整数旋转数通过米尔斯量子化方法和精确WKB与量子化数相关联。 利用最近关于量子力学中精确WKB方法的结果，我们讨论了环面上动力系统和参量振荡器中可能的非微扰效应。 半经典WKB在慢-快动力系统的框架内进行解释。 共伴随Virasoro轨道的分类与Hill方程之间的联系导致了参数空间中相位锁定域的分类，这取决于Virasoro轨道的分类。 我们的观点得到了约瑟夫森结模型和Mathieu方程的数值模拟的支持。 显示英文摘要

摘要： In this study, we discuss the Prufer transform that connects the dynamical system on the torus and the Hill equation, which is interpreted as either the equation of motion for the parametric oscillator or the Schrodinger equation with periodic potential. The structure of phase-locking domains in the dynamical system on torus is mapped into the band-gap structure of the Hill equation. For the parametric oscillator, we provide the relation between the non-adiabatic Hannay angle and the Poincare rotation number of the corresponding dynamical system. In terms of quantum mechanics, the integer rotation number is connected to the quantization number via the Milne quantization approach and exact WKB. Using recent results concerning the exact WKB approach in quantum mechanics, we discuss the possible non-perturbative effects in the dynamical systems on the torus and for parametric oscillator. The semiclassical WKB is interpreted in the framework of a slow-fast dynamical system. The link between the classification of the coadjoint Virasoro orbits and the Hill equation yields a classification of the phase-locking domains in the parameter space in terms of the classification of Virasoro orbits. Our picture is supported by numerical simulations for the model of the Josephson junction and Mathieu equation.