31 research outputs found

    Construction of angular-dependent potentials from trigonometric Pöschl-Teller systems within the Dunkl formalism

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    We generate solvable cases of the two angular equations resulting from variable separation in the three-dimensional Dunkl-Schrödinger equation expressed in spherical coordinates. It is shown that the Dunkl formalism interrelates these angular equations with trigonometric Pöschl-Teller systems. Based on this interrelation, we use point transformations and Darboux-Crum transformations to construct new solvable cases of the angular equations. Instead of the stationary energy, we use the constants due to the separation of variables as transformation parameters for our Darboux-Crum transformations

    Darboux Transformations for orthogonal differential systems and differential Galois Theory

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    Darboux developed an algebraic mechanism to construct an infinite chain of "integrable" second order differential equations as well as their solutions. After a surprisingly long time, Darboux's results had important features in the analytic context, for instance in quantum mechanics where it provides a convenient framework for Supersymmetric Quantum Mechanics. Today, there are a lot of papers regarding the use of Darboux transformations in various contexts, not only in mathematical physics. In this paper, we develop a generalization of the Darboux transformations for tensor product constructions on linear differential equations or systems. Moreover, we provide explicit Darboux transformations for \sym^2 (\mathrm{SL}(2,\mathbb{C})) systems and, as a consequence, also for so(3,CK)\mathfrak{so}(3, C_K) systems, to construct an infinite chain of integrable (in Galois sense) linear differential systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant potentials.Comment: 22 page

    Mathematical methods of factorization and a feedback approach for biological systems

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    The first part of the thesis is devoted to factorizations of linear and nonlinear differential equations leading to solutions of the kink type. The second part contains a study of the synchronization of the chaotic dynamics of two Hodgkin-Huxley neurons by means of the mathematical tools belonging to the geometrical control theory.Comment: Ph. D. Thesis at IPICyT, San Luis Potosi, Mexico, 102 pp, 40 figs. Supervisors: Dr. H.C. Rosu and Dr. R. Fema

    Construction of angular-dependent potentials from trigonometric Pöschl-Teller systems within the Dunkl formalism

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    We generate solvable cases of the two angular equations resulting from variable separation in the three-dimensional Dunkl-Schrödinger equation expressed in spherical coordinates. It is shown that the Dunkl formalism interrelates these angular equations with trigonometric Pöschl-Teller systems. Based on this interrelation, we use point transformations and Darboux-Crum transformations to construct new solvable cases of the angular equations. Instead of the stationary energy, we use the constants due to the separation of variables as transformation parameters for our Darboux-Crum transformations

    Interplay between Riccati, Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

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    Producción CientíficaNonlinear Riccati and Ermakov equations are combined to pair the energy spectrum of 2 different quantum systems via the Darboux method. One of the systems is assumed Hermitian, exactly solvable, with discrete energies in its spectrum. The other system is characterized by a complex‐valued potential that inherits all the energies of the former one and includes an additional real eigenvalue in its discrete spectrum. If such eigenvalue coincides with any discrete energy (or it is located between 2 discrete energies) of the initial system, its presence produces no singularities in the complex‐valued potential. Non‐Hermitian systems with spectrum that includes all the energies of either Morse or trigonometric Pöschl‐Teller potentials are introduced as concrete examples.2019-06-06Ministerio de Economía, Industria y Competitividad (Project MTM2014-57129-C2-1-P)Junta de Castilla y León (programa de apoyo a proyectos de investigación - Ref. VA057U16)CONACyT Scholarships. Grant Numbers: 45454, 48985

    Optical Supersymmetry in the Time Domain

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    Originally emerged within the context of string and quantum field theory, and later fruitfully extrapolated to photonics, the algebraic transformations of quantum-mechanical supersymmetry were conceived in the space realm. Here, we introduce a paradigm shift, demonstrating that Maxwell's equations also possess an underlying supersymmetry in the time domain. As a result, we obtain a simple analytic relation between the scattering coefficients of a large variety of time-varying optical systems and uncover a wide new class of reflectionless, three dimensional, all-dielectric, isotropic, omnidirectional, polarization-independent, non-complex media. Temporal supersymmetry is also shown to arise in dispersive media supporting temporal bound states, which allows engineering their momentum spectra and dispersive properties. These unprecedented features define a promising design platform for free-space and integrated photonics, enabling the creation of a number of novel reconfigurable reflectionless devices, such as frequency-selective, polarization-independent and omnidirectional invisible materials, compact frequency-independent phase shifters, broadband isolators, and versatile pulse-shape transformers

    Symmetries in Quantum Mechanics and Statistical Physics

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    This book collects contributions to the Special Issue entitled "Symmetries in Quantum Mechanics and Statistical Physics" of the journal Symmetry. These contributions focus on recent advancements in the study of PT–invariance of non-Hermitian Hamiltonians, the supersymmetric quantum mechanics of relativistic and non-relativisitc systems, duality transformations for power–law potentials and conformal transformations. New aspects on the spreading of wave packets are also discussed
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