Leaky modes of waveguides as a classical optics analogy of quantum resonances

A classical optics waveguide structure is proposed to simulate resonances of short range one-dimensional potentials in quantum mechanics. The analogy is based on the well known resemblance between the guided and radiation modes of a waveguide with the bound and scattering states of a quantum well. As resonances are scattering states that spend some time in the zone of influence of the scatterer, we associate them with the leaky modes of a waveguide, the latter characterized by suffering attenuation in the direction of propagation but increasing exponentially in the transverse directions. The resemblance is complete since resonances (leaky modes) can be interpreted as bound states (guided modes) with definite lifetime (longitudinal shift). As an immediate application we calculate the leaky modes (resonances) associated with a dielectric homogeneous slab (square well potential) and show that these modes are attenuated as they propagate.Comment: The title has been modified to describe better the contents of the article. Some paragraphs have been added to clarify the result

Superpositions of bright and dark solitons supporting the creation of balanced gain and loss optical potentials

Bright and dark solitons of the cubic nonlinear Schrodinger equation are used to construct complex-valued potentials with all-real spectrum. The real part of these potentials is equal to the intensity of a bright soliton while their imaginary part is defined by the product of such soliton with its concomitant, a dark soliton. Considering light propagation in Kerr media, the real part of the potential refers to the self-focusing of the signal and the imaginary one provides the system with balanced gain-and-loss rates.Comment: 6 figures, 17 pages, LaTeX file. The manuscript has been re-organized (abstract, introduction and conclusions rewritten), and it now includes an appendix with detailed calculations of some relevant results reported in the paper. New references were adde

Dynamical Equations, Invariants and Spectrum Generating Algebras of Mechanical Systems with Position-Dependent Mass

We analyze the dynamical equations obeyed by a classical system with position-dependent mass. It is shown that there is a non-conservative force quadratic in the velocity associated to the variable mass. We construct the Lagrangian and the Hamiltonian for this system and find the modifications required in the Euler-Lagrange and Hamilton's equations to reproduce the appropriate Newton's dynamical law. Since the Hamiltonian is not time invariant, we get a constant of motion suited to write the dynamical equations in the form of the Hamilton's ones. The time-dependent first integrals of motion are then obtained from the factorization of such a constant. A canonical transformation is found to map the variable mass equations to those of a constant mass. As particular cases, we recover some recent results for which the dependence of the mass on the position was already unnoticed, and find new solvable potentials of the P\"oschl-Teller form which seem to be new. The latter are associated to either the su(1,1) or the su(2) Lie algebras depending on the sign of the Hamiltonian

Weighted norm inequalities for the bilinear maximal operator on variable Lebesgue spaces

We extend the theory of weighted norm inequalities on variable Lebesgue spaces to the case of bilinear operators. We introduce a bilinear version of the variable \A_\pp condition, and show that it is necessary and sufficient for the bilinear maximal operator to satisfy a weighted norm inequality. Our work generalizes the linear results of the first author, Fiorenza and Neugebauer \cite{dcu-f-nPreprint2010} in the variable Lebesgue spaces and the bilinear results of Lerner {\em et al.} \cite{MR2483720} in the classical Lebesgue spaces. As an application we prove weighted norm inequalities for bilinear singular integral operators in the variable Lebesgue spaces.Comment: Revised based on anonymous referee's reports. A number of typos and small errors corrected. One conjecture added to introductio

Embeddings between grand, small and variable Lebesgue spaces

We give conditions on the exponent function $p(\cdot)$ that imply the existence of embeddings between grand, small and variable Lebesgue spaces. We construct examples to show that our results are close to optimal. Our work extends recent results by the second author, Rakotoson and Sbordone.Comment: Final version to appear in Math. Note

Completeness and Nonclassicality of Coherent States for Generalized Oscillator Algebras

The purposes of this work are (1) to show that the appropriate generalizations of the oscillator algebra permit the construction of a wide set of nonlinear coherent states in unified form; and (2) to clarify the likely contradiction between the nonclassical properties of such nonlinear coherent states and the possibility of finding a classical analog for them since they are P-represented by a delta function. In (1) we prove that a class of nonlinear coherent states can be constructed to satisfy a closure relation that is expressed uniquely in terms of the Meijer G-function. This property automatically defines the delta distribution as the P-representation of such states. Then, in principle, there must be a classical analog for them. Among other examples, we construct a family of nonlinear coherent states for a representation of the su(1,1) Lie algebra that is realized as a deformation of the oscillator algebra. In (2), we use a beam splitter to show that the nonlinear coherent states exhibit properties like anti-bunching that prohibit a classical description for them. We also show that these states lack second order coherence. That is, although the P-representation of the nonlinear coherent states is a delta function, they are not full coherent. Therefore, the systems associated with the generalized oscillator algebras cannot be considered `classical' in the context of the quantum theory of optical coherence.Comment: 26 pages, 10 figures, minor changes, misprints correcte

Military victims in Colombia and their right to full compensation

La Ley 1448 de 2011 en el Parágrafo 1° del artículo 3, dispone que cuando los miembros de la Fuerza Pública sean víctimas en los términos definidos en mencionado artículo su reparación económica corresponderá por todo concepto al que tengan derecho de acuerdo al régimen especial que les sea aplicable. Ante tal decisión, resulta fundamental entrar a evaluar la pertinencia de esta norma frente al derecho a la igualdad y a la denominada indemnización “a forfait” que surge de la esfera prestacional. Y si la afectación de los derechos a la vida y a la integridad personal a los miembros de la fuerza Pública como consecuencia de graves violaciones al DIH por parte grupos ilegales constituye un riesgo propio de la actividad militar.Law 1448 of 2011 in Paragraph 1 of Article 3, provides that when members of the security forces are victims in the terms defined in that article the financial compensation correspond in every respect to which they are entitled under the regime special to them applicable. Given this decision, it is essential to enter assess the relevance of this rule against the right to equality and the so-called compensation "lift pass" arising from the prestacional sphere. And if the allocation of the rights to life and personal integrity to members of the security forces as a result of serious violations of international humanitarian law by illegal groups constitutes a hazard inherent in military activity

Active Reflection Absorption for a Three Dimensional Multidirectional Wave Generator

In order to implement an accurate system that allows for absorption of reflected waves impinging to a wave maker (Active Reflection Absorption), it was required to apply a method to estimate properly the direction of arrival of the waves that does it in the fastest way possible. Our wavemaker control system has been prepared to handle an algorithm provided by Bosch-Rexroth where the wave angle estimation is practically locked to a very narrow frequency band (spatial gain-mixer). The system was evaluated with physical tests in a 3D wave basin for different conditions of reflected waves arriving with an angle to the wavemaker front, and acceptable performance has been found for the 3D ARA mode. However, for certain conditions over-compensation or sub-compensation can develop resulting in a poor absorption. This is mainly related to not being able to determine accurately the direction from which the reflected waves travel towards the wavemaker. The present work employed concepts found in the areas of antenna array signal processing and signal propagation, which were applied to this problem. This approach coupled naturally with our wavemaker system since it was prepared with 48 gages that can be employed in an array antenna fashion. A program was codified from an algorithm found in literature to calculate the Direction of Arrival (DOA) of the reflected waves. The focus for the testing of this program was with regular waves. The tests were conducted to validate the program with different angles of incidence and show that for regular waves the program was able to detect accurately the DOA of these in as few as 5 snapshots, with a minimum of 7 gages used as the antenna input. With data obtained directly from the control system of our wavemaker using regular waves, the program was able to determine the DOA. The computational burden of the algorithm is not significant in the case of regular waves. A modification of the program is required to analyze the DOA of reflected irregular waves, which could increase the computational burden. Actual implementation of this program to our control system depends on cooperation with Bosch-Rexroth