Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values of the parameter. This is in contrast with what happens in the constant area problem, where the disk is the maximiser only for small values of the boundary parameter. We also present sharp upper and lower bounds for the first eigenvalue of the ball and spherical shells. These results are complemented by the numerical optimisation of the first four and two eigenvalues in 2 and 3 dimensions, respectively, and an evaluation of the quality of the upper bounds obtained. We also study the bifurcations from the ball as the boundary parameter becomes large (negative).Comment: 26 pages, 20 figure

Extinction in the Coma of Comet 17P/Holmes

On 2007 October 29 the outbursting comet 17P/Holmes passed within 0.79 arcsec of a background star. We recorded the event using optical, narrowband photometry and detect a 3% to 4% dip in stellar brightness bracketing the time of closest approach to the comet nucleus. The detected dimming implies an optical depth tau~0.04 at 1.5 arcsec from the nucleus and an optical depth towards the nucleus center tau_n<13.3. At the time of our observations, the coma was optically thick only within rho<~0.01 arcsec from the nucleus. By combining the measured extinction and the scattered light from the coma we estimate a dust red geometric albedo p_d=0.006+/-0.002 at 16 deg phase angle. Our measurements place the most stringent constraints on the extinction optical depth of any cometary coma.Comment: 5 pages, 3 figures, 2 table. Accepted for publication in ApJ

Instability results for the damped wave equation in unbounded domains

We extend some previous results for the damped wave equation in bounded domains in Euclidean spaces to the unbounded case. In particular, we show that if the damping term is of the form $\alpha a$ with bounded $a$ taking on negative values on a set of positive measure, then there will always exist unbounded solutions for sufficiently large positive $\alpha$. In order to prove these results, we generalize some existing results on the asymptotic behaviour of eigencurves of one-parameter families of Schrodinger operators to the unbounded case, which we believe to be of interest in their own right.Comment: LaTeX, 19 pages; to appear in J. Differential Equation

Physics of quantum light emitters in disordered photonic nanostructures

Nanophotonics focuses on the control of light and the interaction with matter by the aid of intricate nanostructures. Typically, a photonic nanostructure is carefully designed for a specific application and any imperfections may reduce its performance, i.e., a thorough investigation of the role of unavoidable fabrication imperfections is essential for any application. However, another approach to nanophotonic applications exists where fabrication disorder is used to induce functionalities by enhancing light-matter interaction. Disorder leads to multiple scattering of light, which is the realm of statistical optics where light propagation requires a statistical description. We review here the recent progress on disordered photonic nanostructures and the potential implications for quantum photonics devices.Comment: Review accepted for publication in Annalen der Physi