Defect Modes and Homogenization of Periodic Schr\"odinger Operators

We consider the discrete eigenvalues of the operator H_\eps=-\Delta+V(\x)+\eps^2Q(\eps\x), where V(\x) is periodic and Q(\y) is localized on $\R^d,\ \ d\ge1$. For \eps>0 and sufficiently small, discrete eigenvalues may bifurcate (emerge) from spectral band edges of the periodic Schr\"odinger operator, H_0 = -\Delta_\x+V(\x), into spectral gaps. The nature of the bifurcation depends on the homogenized Schr\"odinger operator L_{A,Q}=-\nabla_\y\cdot A \nabla_\y +\ Q(\y). Here, $A$ denotes the inverse effective mass matrix, associated with the spectral band edge, which is the site of the bifurcation.Comment: 26 pages, 3 figures, to appear SIAM J. Math. Ana

The nonrelativistic limit of the Magueijo-Smolin model of deformed special relativity

We study the nonrelativistic limit of the motion of a classical particle in a model of deformed special relativity and of the corresponding generalized Klein-Gordon and Dirac equations, and show that they reproduce nonrelativistic classical and quantum mechanics, respectively, although the rest mass of a particle no longer coincides with its inertial mass. This fact clarifies the meaning of the different definitions of velocity of a particle available in DSR literature. Moreover, the rest mass of particles and antiparticles differ, breaking the CPT invariance. This effect is close to observational limits and future experiments may give indications on its effective existence.Comment: 10 pages, plain TeX. Discussion of generalized Dirac equation and CPT violation adde

Grid Voltages Estimation for Three-Phase PWM Rectifiers Control Without AC Voltage Sensors

This paper proposes a new ac voltage sensorless control scheme for the three-phase pulse-width modulation rectifier. A new startup process to ensure a smooth starting of the system is also proposed. The sensorless control scheme uses an adaptive neural (AN) estimator inserted in voltage-oriented control to eliminate the grid voltage sensors. The developed AN estimator combines an AN network in series with an AN filter. The AN estimator structure leads to simple, accurate, and fast grid voltages estimation, and makes it ideal for low-cost digital signal processor implementation. Lyapunov-based stability and parameters tuning of the AN estimator are performed. Simulation and experimental tests are carried out to verify the feasibility and effectiveness of the AN estimator. Obtained results show that the proposed AN estimator presented faster convergence and better accuracy than the second-order generalized integrator-based estimator; the new startup procedure avoided the overcurrent and reduced the settling time; and the AN estimator presented high performances even under distorted and unbalanced grid voltages

Semiclassical Asymptotics for the Maxwell - Dirac System

We study the coupled system of Maxwell and Dirac equations from a semiclassical point of view. A rigorous nonlinear WKB-analysis, locally in time, for solutions of (critical) order $O(\sqrt{\epsilon})$ is performed, where the small semiclassical parameter $\epsilon$ denotes the microscopic/macroscopic scale ratio

A time-splitting spectral scheme for the Maxwell-Dirac system

We present a time-splitting spectral scheme for the Maxwell-Dirac system and similar time-splitting methods for the corresponding asymptotic problems in the semi-classical and the non-relativistic regimes. The scheme for the Maxwell-Dirac system conserves the Lorentz gauge condition, is unconditionally stable and highly efficient as our numerical examples show. In particular we focus in our examples on the creation of positronic modes in the semi-classical regime and on the electron-positron interaction in the non-relativistic regime. Furthermore, in the non-relativistic regime, our numerical method exhibits uniform convergence in the small parameter \dt, which is the ratio of the characteristic speed and the speed of light.Comment: 29 pages, 119 figure

Semi- and Non-relativistic Limit of the Dirac Dynamics with External Fields

We show how to approximate Dirac dynamics for electronic initial states by semi- and non-relativistic dynamics. To leading order, these are generated by the semi- and non-relativistic Pauli hamiltonian where the kinetic energy is related to $\sqrt{m^2 + \xi^2}$ and $\xi^2 / 2m$, respectively. Higher-order corrections can in principle be computed to any order in the small parameter v/c which is the ratio of typical speeds to the speed of light. Our results imply the dynamics for electronic and positronic states decouple to any order in v/c << 1. To decide whether to get semi- or non-relativistic effective dynamics, one needs to choose a scaling for the kinetic momentum operator. Then the effective dynamics are derived using space-adiabatic perturbation theory by Panati et. al with the novel input of a magnetic pseudodifferential calculus adapted to either the semi- or non-relativistic scaling.Comment: 42 page