We present an interpretation of the functions appearing in the Wei–Norman factorization of the evolution operator for a Hamiltonian belonging to the SU(1,1) algebra in terms of the classical solutions of the Generalized Caldirola–Kanai (GCK) oscillator (with time-dependent mass and frequency). Choosing P2, X2, and the dilation operator as a basis for the Lie algebra, we obtain that, out of the six possible orderings for the Wei–Norman factorization of the evolution operator for the GCK Hamiltonian, three of them can be expressed in terms of its classical solutions and the other three involve the classical solutions associated with a mirror Hamiltonian obtained by inverting the mass. In addition, we generalize the Wei–Norman procedure to compute the factorization of other operators, such as a generalized Fresnel transform and the Arnold transform (and its generalizations), obtaining also in these cases a semiclassical interpretation for the functions in the exponents of the Wei–Norman factorization. The singularities of the functions appearing in the Wei–Norman factorization are related to the caustic points of Morse theory, and the expression of the evolution operator at the caustics is obtained using a limiting procedure, where the Fourier transform of the initial state appears along with the Guoy phase.

中文翻译：

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SU(1, 1) 的 Wei-Norman 分解及其相关积分变换的半经典解释

我们根据广义 Caldirola-Kanai (GCK) 振荡器的经典解（随时间变化）对属于 SU(1,1) 代数的哈密顿量的演化算子的​​ Wei-Norman 分解中出现的函数进行解释-依赖的质量和频率）。选择 P2、X2 和膨胀算子作为李代数的基础，我们得到，在 GCK 哈密顿量演化算子的​​ Wei-Norman 分解的六种可能排序中，其中三种可以表示为其经典解的一部分，其他三个涉及与通过反转质量获得的镜像哈密顿量相关的经典解。此外，我们推广了 Wei-Norman 过程来计算其他运算符的分解，例如广义菲涅尔变换和阿诺德变换（及其推广），在这些情况下也获得对 Wei-Norman 分解指数中函数的半经典解释。Wei-Norman 分解中出现的函数的奇异性与 Morse 理论的焦散点有关，焦散处的演化算子的​​表达式是使用限制程序获得的，其中初始状态的傅立叶变换出现在郭相。