标题： 精确Ermakov-Pinney解的构造与时间相关的量子振荡器 显示英文标题

标题： Construction of Exact Ermakov-Pinney Solutions and Time-Dependent Quantum Oscillators

摘要： 具有时变频率的谐振子对于含时薛定谔方程有一族线性量子不变量，这些不变量由经典运动方程的任意两个独立解确定。 埃尔马克夫（Ermakov）和平尼（Pinney）已经证明，含时振荡器中带反立方项的一般解可以用含时振荡器的两个独立解来表示。 我们探讨了含时谐振子的线性量子不变量与埃尔马克夫-平尼解之间的联系。 我们提出了一种新颖的方法来构造一类含时振荡器的埃尔马克夫-平尼解以及含时薛定谔方程的波函数。 我们进一步表明，第一类和第二类波施尔-泰勒（Pöschl-Teller）势属于一类特殊的精确含时振荡器。 提出了一种适用于任何缓慢变化的时变频率的微扰方法。 显示英文摘要

摘要： The harmonic oscillator with a time-dependent frequency has a family of linear quantum invariants for the time-dependent Schr\"{o}dinger equation, which are determined by any two independent solutions to the classical equation of motion. Ermakov and Pinney have shown that a general solution to the time-dependent oscillator with an inverse cubic term can be expressed in terms of two independent solutions to the time-dependent oscillator. We explore the connection between linear quantum invariants and the Ermakov-Pinney solution for the time-dependent harmonic oscillator. We advance a novel method to construct Ermakov-Pinney solutions to a class of time-dependent oscillators and the wave functions for the time-dependent Schr\"{o}dinger equation. We further show that the first and the second P\"{o}schl-Teller potentials belong to a special class of exact time-dependent oscillators. A perturbation method is proposed for any slowly-varying time-dependent frequency.