Semiclassical interpretation of Wei–Norman factorization for SU(1,1) and its related integral transforms

Abstract

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.