31 research outputs found
Construction of angular-dependent potentials from trigonometric Pöschl-Teller systems within the Dunkl formalism
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
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 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
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
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
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
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
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Nonlinear classical and quantum integrable systems with PT -symmetries
A key feature of integrable systems is that they can be solved to obtain exact analytical solutions. In this thesis we show how new models can be found through generalisations of some well known nonlinear partial differential equations including the Korteweg-de Vries, modified Korteweg-de Vries, sine-Gordon, Hirota, Heisenberg and Landau-Lifschitz types with joint parity and time symmetries whilst preserving integrability properties.
The first joint parity and time symmetric generalizations we take are extensions to the complex and multicomplex fields, such as bicomplex, quaternionic, coquaternionic and octonionic types. Subsequently, we develop new methods from well-known ones, such as Hirota’s direct method, Bäcklund transformations and Darboux-Crum transformations to solve for these newsystems to obtain exact analytical solutions of soliton and multi-soliton types. Moreover, in agreement with the reality property present in joint parity and time symmetric non-Hermitian quantum systems, we find joint parity and time symmetries also play a key role for reality of conserved charges for the new systems, even though the soliton solutions are complex or multicomplex.
Our complex extensions have proved to be successful in helping one to obtain regularized degenerate multi-soliton solutions for the Korteweg-de Vries equation, which has not been realised before. We extend our investigations to explore degenerate multi-soliton solutions for the sine-Gordon equation and Hirota equation. In particular, we find the usual time-delays from degenerate soliton solution scattering are time-dependent, unlike the non-degenerate multi-soliton solutions, and provide a universal formula to compute the exact time-delay values for scattering of N-soliton solutions.
Other joint parity and time symmetric extensions of integrable systems we take are of nonlocal nature, with nonlocalities in space and/or in time, of time crystal type. Whilst developing new methods for the construction of soliton solutions for these systems, we xiv find new types of solutions with different parameter dependence and qualitative behaviour even in the one-soliton solution cases. We exploit gauge equivalence between the Hirota system with continuous Heisenberg and Landau-Lifschitz systems to see how nonlocality is inherited from one system to another and vice versa. In the final part of the thesis, we extend some of our investigations to the quantum regime. In particularwe generalize the scheme of Darboux transformations for fully timedependent non-Hermitian quantum systems, which allows us to create an infinite tower of solvable models
Symmetries in Quantum Mechanics and Statistical Physics
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