Semiclassical trajectory-coherent approximation in quantum mechanics I. High-order corrections to multidimensional time-dependent equations of Schrödinger type

A concept of semiclassically concentrated states is developed on the basis of the Maslov germ theory. Higher approximations of semiclassical trajectory-coherent states and of semiclassical Green function (in the class of semiclassically concentrated states) for a many-dimensional Schrödinger-type equation are constructed. It is shown that, in class of such semiclassically concentrated states, a Schrödinger-type equation (up to any order of ℏ, ℏ → 0) is equivalent, from the viewpoint of calculating the quantum averages, to a closed finite system of ordinary differential equations.