Perturbation Theory for PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold

The $PT$ symmetric potential $V_0[\cos(2\pi x/a)+i\lambda\sin(2\pi x/a)]$ has a completely real spectrum for $\lambda\le 1$, and begins to develop complex eigenvalues for $\lambda>1$. At the symmetry-breaking threshold $\lambda=1$ some of the eigenvectors become degenerate, giving rise to a Jordan-block structure for each degenerate eigenvector. In general this is expected to give rise to a secular growth in the amplitude of the wave. However, it has been shown in a recent paper by Longhi, by numerical simulation and by the use of perturbation theory, that for an initial wave packet this growth is suppressed, giving instead a constant maximum amplitude. We revisit this problem by developing the perturbation theory further. We verify that the results found by Longhi persist to second order, and with different input wave packets we are able to see the seeds in perturbation theory of the phenomenon of birefringence first discovered by Makris et al.Comment: Some references correcte

First order structure-preserving perturbation theory for eigenvalues of symplectic matrices

A first order perturbation theory for eigenvalues of real or complex J-symplectic matrices under structure- preserving perturbations is developed. As main tools structured canonical forms and Lidskii-like formulas for eigenvalues of multiplicative perturbations are used. Explicit formulas, depending only on appropriately normalized left and right eigenvectors, are obtained for the leading terms of asymptotic expansions describing the perturbed eigenvalues. Special attention is given to eigenvalues on the unit circle, especially to the exceptional eigenvalues ±1, whose behavior under structure-preserving perturbations is known to differ significantly from the behavior under general perturbations. Several numerical examples are used to illustrate the asymptotic expansions

Long-wavelength metric backreactions in slow-roll inflation

We examine the importance of second order corrections to linearized cosmological perturbation theory in an inflationary background, taken to be a spatially flat FRW spacetime. The full second order problem is solved in the sense that we evaluate the effect of the superhorizon second order corrections on the inhomogeneous and homogeneous modes of the linearized flucuations. These second order corrections enter in the form of a {\it cumulative} contribution from {\it all} of their Fourier modes. In order to quantify their physical significance we study their effective equation of state by looking at the perturbed energy density and isotropic pressure to second order. We define the energy density (isotropic pressure) in terms of the (averaged) eigenvalues associated with timelike (spacelike) eigenvectors of a total stress energy for the metric and matter fluctuations. Our work suggests that that for many parameters of slow-roll inflation, the second order contributions to these energy density and pressures may dominate over the first order effects for the case of super-Hubble evolution. These results hold in our choice of first and second order coordinate conditions however we also argue that other `reasonable` coordinate conditions do not alter the relative importance of the second order terms. We find that these second order contributions approximately take the form of a cosmological constant in this coordinate gauge, as found by others using effective methods.Comment: Submitted to Phys. Rev.