The Anisotropic Gaussian Semi-Classical Schrödinger Propagator

We present a construction of the anisotropic Gaussian semi-classical Schrödinger propagator, emblematic of a class of Fourier integral operators of quadratic phase kernels related to the Schrödinger equation. We deduce a set of algebraic relations of the variational matrices, solutions of the variational system pertaining to single Gaussian wave packet semi-classical time evolution, some already known in the literature, representing the symplectic and other invariances of the dynamics, which are subsequently utilized in order to derive the Van Vleck formula from the semi-classical Schrödinger propagator.