Dynamics and localization of dressed Schrodinger and Relativistic Dirac electron states in graphene

We have investigated the dynamics of single particle in a space-localized but periodically varying time-dependent potential $V(x,t)=\pm V_0 a \delta(x)\cos(\omega t)$ with frequency $\omega$.
This potential may be applied to either Schr\"odinger or relativistic Dirac particles. Both numerical and semi-analytic solutions have been obtained for specific initial conditions for the probability
distribution of the particle originally around a localization center. For negative potential, we solve a bound-state problem, whereas for positive potential, the particle is scattered from a time-dependent potential barrier. From a physical point of view, this problem is relevant to space-localized electron-dressed states for a Dirac particle interacting with linearly-polarized light.