Half eigenvalues and the Fucik spectrum of multi-point, boundary value problems

We consider the nonlinear boundary value problem consisting of the equation \tag{1} -u" = f(u) + h, \quad \text{a.e. on $(-1,1)$,} where $h \in L^1(-1,1)$, together with the multi-point, Dirichlet-type boundary conditions \tag{2} u(\pm 1) = \sum^{m^\pm}_{i=1}\alpha^\pm_i u(\eta^\pm_i) where $m^\pm \ge 1$ are integers, $\alpha^\pm = (\alpha_1^\pm, ...,\alpha_m^\pm) \in [0,1)^{m^\pm}$, $\eta^\pm \in (-1,1)^{m^\pm}$, and we suppose that $$\sum_{i=1}^{m^\pm} \alpha_i^\pm < 1 .$$ We also suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous, and $$0 < f_{\pm\infty}:=\lim_{s \to \pm\infty} \frac{f(s)}{s} < \infty.$$ We allow $f_{\infty} \ne f_{-\infty}$ --- such a nonlinearity $f$ is {\em jumping}. Related to (1) is the equation \tag{3} -u" = \lambda(a u^+ - b u^-), \quad \text{on $(-1,1)$,} where $\lambda,\,a,\,b > 0$, and $u^{\pm}(x) =\max\{\pm u(x),0\}$ for $x \in [-1,1]$. The problem (2)-(3) is `positively-homogeneous' and jumping. Regarding $a,\,b$ as fixed, values of $\lambda = \lambda(a,b)$ for which (2)-(3) has a non-trivial solution $u$ will be called {\em half-eigenvalues}, while the corresponding solutions $u$ will be called {\em half-eigenfunctions}. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having specified nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to solvability and non-solvability results for the problem (1)-(2). The set of half-eigenvalues is closely related to the `Fucik spectrum' of the problem, which we briefly describe. Equivalent solvability and non-solvability results for (1)-(2) are obtained from either the half-eigenvalue or the Fucik spectrum approach

Second order, multi-point problems with variable coefficients

In this paper we consider the eigenvalue problem consisting of the equation -u" = \la r u, \quad \text{on $(-1,1)$}, where $r \in C^1[-1,1], \ r>0$ and \la \in \R, together with the multi-point boundary conditions u(\pm 1) = \sum^{m^\pm}_{i=1} \al^\pm_i u(\eta^\pm_i), where $m^\pm \ge 1$ are integers, and, for $i = 1,...,m^\pm$, \al_i^\pm \in \R, $\eta_i^\pm \in [-1,1]$, with $\eta_i^+ \ne 1$, $\eta_i^- \ne -1$. We show that if the coefficients \al_i^\pm \in \R are sufficiently small (depending on $r$) then the spectral properties of this problem are similar to those of the usual separated problem, but if the coefficients \al_i^\pm are not sufficiently small then these standard spectral properties need not hold. The spectral properties of such multi-point problems have been obtained before for the constant coefficient case ($r \equiv 1$), but the variable coefficient case has not been considered previously (apart from the existence of `principal' eigenvalues). Some nonlinear multi-point problems are also considered. We obtain a (partial) Rabinowitz-type result on global bifurcation from the eigenvalues, and various nonresonance conditions for existence of general solutions and also of nodal solutions --- these results rely on the spectral properties of the linear problem

Instability of equatorial water waves with an underlying current

In this paper we use the short-wavelength instability approach to derive an instability threshold for exact trapped equatorial waves propagating eastwards in the presence of an underlying current

Coherent Excitonic Coupling in an Asymmetric Double InGaAs Quantum Well Arises from Many-Body Effects

We study an asymmetric double InGaAs quantum well using optical two-dimensional coherent spectroscopy. The collection of zero-quantum, one-quantum, and two-quantum two-dimensional spectra provides a unique and comprehensive picture of the double well coherent optical response. Coherent and incoherent contributions to the coupling between the two quantum well excitons are clearly separated. An excellent agreement with density matrix calculations reveals that coherent interwell coupling originates from many-body interactions

Polariton quantum boxes in semiconductor microcavities

We report on the realization of polariton quantum boxes in a semiconductor microcavity under strong coupling regime. The quantum boxes consist of mesas that confine the cavity photon, etched on top of the spacer of a microcavity. For mesas with sizes of the order of a few micron in width and nm in depth, we observe quantization, caused by the lateral confinement, of the polariton modes in several peaks. We evidence the strong exciton-photon coupling regime through a typical/clear anticrossing curve for each quantized level. Moreover the growth technique is of high quality, which opens the way for the conception of new optoelectronic devices

EU-Raw Materials Intelligence Capacity Platform (EU-RMCP) – Technical system specification

EU-Raw Materials Intelligence Capacity Platform (or EU-RMICP) integrates metadata on data sources related to primary and secondary mineral resources and brings the end users an expertise on the methods and tools used in mineral intelligence. The system is capable of bringing relevant user ‘answers’ of the type 'how to proceed for …' on almost any question related to mineral resources, on the whole supply chain, from prospecting to recycling, taking into account the environmental, political and social dimensions. EU-RMICP is based on an ontology of the domain of mineral resources (coupled with more generic cross-functional ontologies, relative to commodities, time and space), which represents the domain of the questions of the users (experts and non-experts). The user navigates in the ontology by using a Dynamic Graph of Decision (DDG), which allows him/her to discover the solutions which he/she is looking for without having to formulate any question. The system is coupled with a 'RDF Triple Store' (a database storing the ontologies), factSheets, doc-Sheets and flowSheets (i.e., specific formatted forms) related to methods and documentation, scenarios and metadata.JRC.B.6-Digital Econom

Analytical method for determining quantum well exciton properties in a magnetic field

We develop an analytical approximate method for determining the Bohr radii of Wannier-Mott excitons in thin quantum wells under the influence of magnetic field perpendicular to the quantum well plane. Our hybrid variational-perturbative method allows us to obtain simple closed formulas for exciton binding energies and optical transition rates. We confirm the reliability of our method through exciton-polariton experiments realized in a GaAs/AlAs microcavity with an 8 nm In-x Ga1-xAs quantum well and magnetic field strengths as high as 14 T

Resonant nonlinear studies of trapped 0D-microcavity polaritons

We performed studies on microcavity polaritons trapped along the three dimensions of space, under resonant excitation on a confined lower polariton state. We observed various nonlinear behaviors as a function of the pump power, without any apparent loss of the strong-coupling. That may be understood as effects of Coulomb interaction. Indications of bistable behaviors in the system are observed and discussed

Nonlinear relaxation of zero-dimension-trapped microcavity polaritons

We study the emission properties of confined polariton states in shallow zero-dimensional traps under nonresonant excitation. We evidence several relaxation regimes. For slightly negative photon-exciton detuning, we observe a nonlinear increase of the emission intensity, characteristic of carrier-carrier scattering assisted relaxation under strong-coupling regime. This demonstrates the efficient relaxation toward a confined state of the system. For slightly positive detuning, we observe the transition from strong to weak coupling regime and then to single-mode lasing

Three-Dimensional Multiple-Order Twinning of Self-Catalyzed GaAs Nanowires on Si Substrates

In this paper we introduce a new paradigm for nanowire growth that explains the unwanted appearance of parasitic nonvertical nanowires. With a crystal structure polarization analysis of the initial stages of GaAs nanowire growth on Si substrates, we demonstrate that secondary seeds form due to a three-dimensional twinning phenomenon. We derive the geometrical rules that underlie the multiple growth directions observed experimentally. These rules help optimizing nanowire array devices such as solar or water splitting cells or of more complex hierarchical branched nanowire devices