Floquet theory and continued fractions for harmonically driven systems

We derive an exact solution using continued fractions for a quantum particle scattering from an oscillating delta-function potential. We study its transmission properties such as: Transmission zeros, transmission poles and threshold anomalies. Using the same technique and a translation matrix method, we study the problem of an infinite chain of oscillating deltas. We calculate its band structure and eigenstates and show explicitly the contribution to these eigenstates from the quasi-bound state of a single oscillating delta. We study the dynamics of the quasi-energy bands of the system as a function of the strength of the oscillation and show band quasi-periodicity and band collapse. We also define the Floquet-Green’s function for a time-periodic Hamiltonian and by a generalization of the method used for the two previous potentials we are able to derive an expression for the Floquet-Green’s function of any harmonically driven Hamiltonian. As an example of the application of this method we study a tight-binding Hamiltonian with harmonic time dependence.