Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the one-dimensional stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov--Schmidt reductions in Fourier space. In particular, existence of periodic/anti-periodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.Comment: 18 pages, no figure

Distribution of resonances for open quantum maps

We analyze simple models of classical chaotic open systems and of their quantizations (open quantum maps on the torus). Our models are similar to models recently studied in atomic and mesoscopic physics. They provide a numerical confirmation of the fractal Weyl law for the density of quantum resonances of such systems. The exponent in that law is related to the dimension of the classical repeller (or trapped set) of the system. In a simplified model, a rigorous argument gives the full resonance spectrum, which satisfies the fractal Weyl law. For this model, we can also compute a quantity characterizing the fluctuations of conductance through the system, namely the shot noise power: the value we obtain is close to the prediction of random matrix theory.Comment: 60 pages, no figures (numerical results are shown in other references

Resolvent estimates for normally hyperbolic trapped sets

We give pole free strips and estimates for resolvents of semiclassical operators which, on the level of the classical flow, have normally hyperbolic smooth trapped sets of codimension two in phase space. Such trapped sets are structurally stable and our motivation comes partly from considering the wave equation for Kerr black holes and their perturbations, whose trapped sets have precisely this structure. We give applications including local smoothing effects with epsilon derivative loss for the Schr\"odinger propagator as well as local energy decay results for the wave equation.Comment: Further changes to erratum correcting small problems with Section 3.5 and Lemma 4.1; this now also corrects hypotheses, explicitly requiring trapped set to be symplectic. Erratum follows references in this versio

Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian

We consider the Landau Hamiltonian (i.e. the 2D Schroedinger operator with constant magnetic field) perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the Radon transform of the perturbation potential V.Comment: 30 pages. The explicit dependence on B and V in Theorem 1.6 (i) - (ii) indicated. Typos corrected. To appear in Communications in Mathematical Physic

Wigner's Dynamical Transition State Theory in Phase Space: Classical and Quantum

A quantum version of transition state theory based on a quantum normal form (QNF) expansion about a saddle-centre-...-centre equilibrium point is presented. A general algorithm is provided which allows one to explictly compute QNF to any desired order. This leads to an efficient procedure to compute quantum reaction rates and the associated Gamov-Siegert resonances. In the classical limit the QNF reduces to the classical normal form which leads to the recently developed phase space realisation of Wigner's transition state theory. It is shown that the phase space structures that govern the classical reaction d ynamicsform a skeleton for the quantum scattering and resonance wavefunctions which can also be computed from the QNF. Several examples are worked out explicitly to illustrate the efficiency of the procedure presented.Comment: 132 pages, 31 figures, corrected version, Nonlinearity, 21 (2008) R1-R11

RhoGTPase Regulators Orchestrate Distinct Stages of Synaptic Development

Small RhoGTPases regulate changes in post-synaptic spine morphology and density that support learning and memory. They are also major targets of synaptic disorders, including Autism. Here we sought to determine whether upstream RhoGTPase regulators, including GEFs, GAPs, and GDIs, sculpt specific stages of synaptic development. The majority of examined molecules uniquely regulate either early spine precursor formation or later matura- tion. Specifically, an activator of actin polymerization, the Rac1 GEF β-PIX, drives spine pre- cursor formation, whereas both FRABIN, a Cdc42 GEF, and OLIGOPHRENIN-1, a RhoA GAP, regulate spine precursor elongation. However, in later development, a novel Rac1 GAP, ARHGAP23, and RhoGDIs inactivate actomyosin dynamics to stabilize mature synap- ses. Our observations demonstrate that specific combinations of RhoGTPase regulatory pro- teins temporally balance RhoGTPase activity during post-synaptic spine development