Quasi-periodic states of an oscillatory hamiltonian

The Schrödinger equation for a finite-basis system with an oscillatory hamiltonian has a complete orthonormal set of solutions (quasi-periodic states) which are eigenfunctions of the evolution operator over one period of the hamiltonian. Expectation values evaluated on these states oscillate with the same period as the hamiltonian. Although infinite-basis systems are not expected to have true quasi-periodic states, one can develop perturbation expansions for these states which have a steady-state oscillatory behaviour and which, when appropriately truncated, can be good approximate solutions to the time-dependent Schrödinger equation with an oscillatory hamiltonian. The low-order solutions are compared with an earlier ‘steady-state’ perturbation theory.