Non-autonomous Hamiltonian quantum systems, operator equations and representations of Bender-Dunne Weyl ordered basis under time-dependent canonical transformations

We solve the problem of integrating operator equations for the dynamics of nonautonomous quantum systems by using time-dependent canonical transformations. The studied operator equations essentially reproduce the classical integrability conditions at the quantum level in the basic cases of one-dimensional nonautonomous dynamical systems. We seek solutions in the form of operator series in the Bender–Dunne basis of pseudodifferential operators. Together with this problem, we consider quantum canonical transformations. The minimal solution of the operator equation in the representation of the basis at a fixed time corresponds to the lowest-order contribution of the solution obtained as a result of applying a canonical linear transformation to the basis elements.