Superintegrability of the Fock-Darwin system

The Fock-Darwin system is analysed from the point of view of its symmetry properties in the quantum and classical frameworks. The quantum Fock-Darwin system is known to have two sets of ladder operators, a fact which guarantees its solvability. We show that for rational values of the quotient of two relevant frequencies, this system is superintegrable, the quantum symmetries being responsible for the degeneracy of the energy levels. These symmetries are of higher order and close a polynomial algebra. In the classical case, the ladder operators are replaced by ladder functions and the symmetries by constants of motion. We also prove that the rational classical system is superintegrable and its trajectories are closed. The constants of motion are also generators of symmetry transformations in the phase space that have been integrated for some special cases. These transformations connect different trajectories with the same energy. The coherent states of the quantum superintegrable system are found and they reproduce the closed trajectories of the classical one.Comment: 21 pages,16 figure

Classical ladder functions for Rosen-Morse and curved Kepler-Coulomb systems

Producción CientíficaLadder functions in classical mechanics are defined in a similar way as ladder operators in the context of quantum mechanics. In the present paper, we develop a new method for obtaining ladder functions of one dimensional systems by means of a product of two ‘factor functions’. We apply this method to the curved Kepler–Coulomb and Rosen–Morse II systems whose ladder functions were not found yet. The ladder functions here obtained are applied to get the motion of the systems.Ministerio de Economía, Industria y Competitividad (project MTM2014-57129-C2-1-P)Junta de Castilla y León-FEDER (projects BU229P18 / VA057U16 / VA137G18)

Tavis-Cummings models and their quasi-exactly solvable Schrödinger Hamiltonians

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Heisenberg-type higher order symmetries of superintegrable systems separable in cartesian coordinates

Heisenberg-type higher order symmetries are studied for both classical and quantum mechanical systems separable in cartesian coordinates. A few particular cases of this type of superintegrable systems were already considered in the literature, but here they are characterized in full generality together with their integrability properties. Some of these systems are defined only in a region of $\mathbb R^n$, and in general they do not include bounded solutions. The quantum symmetries and potentials are shown to reduce to their superintegrable classical analogs in the $\hbar \to0$ limit.Comment: 23 Pages, 3 figures, To appear in Nonlinearit

Modelling lava flows by Cellular Nonlinear Networks (CNN): preliminary results

International audienceThe forecasting of lava flow paths is a complex problem in which temperature, rheology and flux-rate all vary with space and time. The problem is more difficult to solve when lava runs down a real topography, considering that the relations between characteristic parameters of flow are typically nonlinear. An alternative approach to this problem that does not use standard differential equation methods is Cellular Nonlinear Networks (CNNs). The CNN paradigm is a natural and flexible framework for describing locally interconnected, simple, dynamic systems that have a lattice-like structure. They consist of arrays of essentially simple, nonlinearly coupled dynamic circuits containing linear and non-linear elements able to process large amounts of information in real time. Two different approaches have been implemented in simulating some lava flows. Firstly, a typical technique of the CNNs to analyze spatio-temporal phenomena (as Autowaves) in 2-D and in 3-D has been utilized. Secondly, the CNNs have been used as solvers of partial differential equations of the Navier-Stokes treatment of Newtonian flow

Improved graphene blisters by ultrahigh pressure sealing

Graphene is a very attractive material for nanomechanical devices and membrane applications. Graphene blisters based on silicon oxide micro-cavities are a simple but relevant example of nanoactuators. A drawback of this experimental set up is that gas leakage through the graphene-SiO2 interface contributes significantly to the total leak rate. Here we study the diffusion of air from pressurized graphene drumheads on SiO2 micro-cavities and propose a straightforward method to improve the already strong adhesion between graphene and the underlying SiO2 substrate, resulting in reduced leak rates. This is carried out by applying controlled and localized ultrahigh pressure (> 10 GPa) with an Atomic Force Microscopy diamond tip. With this procedure, we are able to significantly approach the graphene layer to the SiO2 surface around the drumheads, thus enhancing the interaction between them allowing us to better seal the graphene-SiO2 interface, which is reflected in up to ~ 4 times lower leakage rates. Our work opens an easy way to improve the performance of graphene as a gas membrane on a technological relevant substrate such as SiO2.Comment: pages 19, 4 figures + supplementary informatio