The nonlinear one-dimensional model of controllable photonic crystals is considered. A photonic crystal is formed in a dielectric film on which the electromagnetic pump and signal waves fall. It is shown that the shape of the superlattice is controlled by the variation of the external electrostatic field imposed on the film. Thus it leads to changing boundaries of the allowed and forbidden zone...

We prove a dispersive estimate for the Schrödinger equation on the real line, mapping between weighted L spaces with stronger time-decay (|t|− 3 2 versus |t|− 1 2 ) than is possible on unweighted spaces. To satisfy this bound, the long-term behavior of solutions must include transport away from the origin. Our primary requirements are that 〈x〉V be integrable and −∆+V not have a resonance at zer...

We establish necessary and sufficient conditions for the one-dimensional differential Hardy inequality to hold, including the overdetermined case. The solution is given in terms different from those of the known results. Moreover, the least constant for this inequality is estimated.