Distinguished orbits and the L-S category of simply connected compact Lie groups

We show that the Lusternik-Schnirelmann category of a simple, simply connected, compact Lie group G is bounded above by the sum of the relative categories of certain distinguished conjugacy classes in G corresponding to the vertices of the fundamental alcove for the action of the affine Weyl group on the Lie algebra of a maximal torus of G.Comment: 10 pages, 2 figure

Resonance widths for the molecular predissociation

We consider a semiclassical $2\times 2$ matrix Schr\"odinger operator of the form $P=-h^2\Delta {\bf I}_2 + {\rm diag}(V_1(x), V_2(x)) +hR(x,hD_x)$, where $V_1, V_2$ are real-analytic, $V_2$ admits a non degenerate minimum at 0, $V_1$ is non trapping at energy $V_2(0)=0$, and $R(x,hD_x)=(r_{j,k}(x,hD_x))_{1\leq j,k\leq 2}$ is a symmetric off-diagonal $2\times 2$ matrix of first-order pseudodifferential operators with analytic symbols. We also assume that $V_1(0) >0$. Then, denoting by $e_1$ the first eigenvalue of -\Delta + \la V_2"(0)x,x\ra /2, and under some ellipticity condition on $r_{1,2}$ and additional generic geometric assumptions, we show that the unique resonance $\rho_1$ of $P$ such that $\rho_1 = V_2(0) + (e_1+r_{2,2}(0,0))h + {\mathcal O}(h^2)$ (as $h\rightarrow 0_+$) satisfies, $$\Im \rho_1 = -h^{n_0+(1-n_\Gamma)/2}f(h,\ln\frac1{h})e^{-2S/h},$$ where $f(h,\ln\frac1{h}) \sim \sum_{0\leq m\leq\ell} f_{\ell,m}h^\ell(\ln\frac1{h})^m$ is a symbol with $f_{0,0}>0$, $S>0$ is the so-called Agmon distance associated with the degenerate metric $\max(0, \min(V_1,V_2))dx^2$, between 0 and $\{V_1\leq 0\}$, and $n_0\geq 1$, $n_{\Gamma}\geq 0$ are integers that depend on the geometry.Comment: 37 pages, no figur

Ultrafast WDM logic

Ultrafast all-optical logic gates that accept optical inputs in which wavelength designates bit position within the overall byte are proposed and demonstrated. Four-wave mixing is shown to provide a conditional test function that can be used to construct any multi-input logic gate. Polarization provides the logic state for each bit. Implementations that use semiconductor optical amplifiers as the four-wave mixing medium can be monolithic and compact

Spectral logic gates for byte-wide WDM signal processing

We propose a new class of all-optical logic gates based on four-wave mixing, designed to operate on multiwavelength input signals. We demonstrate the XOR function with two 2.5-Gbit/s modulated input signals