J. Negro

Nonrelativistic conformal groups

In this work a systematic study of finite-dimensional nonrelativistic conformal groups is carried out under two complementary points of view. First, the conformal Killing equation is solved to obtain a whole family of finite-dimensional conformal algebras corresponding to each of the Galilei and Newton–Hooke kinematical groups. Some of their algebraic and geometrical properties are studied in a second step. Among the groups included in these families one can identify, for example, the contraction of the Minkowski conformal group, the analog for a nonrelativistic de Sitter space, or the nonextended Schrodinger group.

Second-order supersymmetric periodic potentials

Abstract Irreducible second-order SUSY transformations are applied to periodic Hamiltonians in order to find physically acceptable partner potentials with the same band structure as the initial one. Lames potentials are analized in the same context. The main differences with the SUSY approach to potentials allowing for a discrete spectrum are also discussed.

Nonrelativistic conformal groups. II. Further developments and physical applications

The finite-dimensional conformal groups associated with the Galilei and (oscillating or expanding) Newton–Hooke space–time manifolds was characterized by the present authors in a recent work. Three isomorphic group families, one for each nonrelativistic kinematics, were obtained, whose members are labeled by a half-integer number l. Since the action of these groups on their corresponding space–time manifolds is only local, a linearization is introduced here such that the corresponding action is well defined everywhere. In particular, the (l=1)-conformal cases that can be obtained by contraction from the well-known Minkowskian conformal group are treated in more detail. As an application of physical interest, the conformal invariance of the Galilean electromagnetism is studied. In order to achieve it, the pertinent local representations of the Galilean conformal algebras are derived.

Exact analytic solutions for a Dirac electron moving in graphene under magnetic fields

Exact analytical solutions for the bound states of a graphene Dirac electron in various magnetic fields with translational symmetry are obtained. In order to solve the time-independent Dirac-Weyl equation the factorization method used in supersymmetric quantum mechanics is adapted to this problem. The behavior of the discrete spectrum, probability and current densities are discussed.

The supersymmetric modified Pöschl-Teller and delta well potentials

New supersymmetric (SUSY) partners of the modified Poschl-Teller and the Diracs delta well potentials are constructed in closed form. The resulting one-parametric potentials are shown to be interrelated by a limiting process. The range of values of the parameters for which these potentials are free of singularities is exactly determined. The construction of higher-order SUSY partner potentials is also investigated.

A study of the bound states for square potential wells with position-dependent mass

A potential well with position-dependent mass is studied for bound states. Applying appropriate matching conditions, a transcendental equation is derived for the energy eigenvalues. Numerical results are presented graphically and the variation of the energy of the bound states are calculated as a function of the well-width and mass.

Polynomial Heisenberg algebras

Polynomial deformations of the Heisenberg algebra are studied in detail. Some of their natural realizations are given by the higher order susy partners (and not only by those of first order, as is already known) of the harmonic oscillator for even-order polynomials. Here, it is shown that the susy partners of the radial oscillator play a similar role when the order of the polynomial is odd. Moreover, it will be proved that the general systems ruled by such kinds of algebras, in the quadratic and cubic cases, involve Painleve transcendents of types IV and V, respectively.

Factorizations of one-dimensional classical systems

Abstract A class of one-dimensional classical systems is characterized from an algebraic point of view. The Hamiltonians of these systems are factorized in terms of two functions that together with the Hamiltonian itself close a Poisson algebra. These two functions lead directly to two time-dependent integrals of motion from which the phase motions are derived algebraically. The systems so obtained constitute the classical analogues of the well known factorizable one-dimensional quantum mechanical systems.

Traveling-Wave Solutions for Korteweg–de Vries–Burgers Equations through Factorizations

Traveling-wave solutions of the standard and compound form of Korteweg–de Vries–Burgers equations are found using factorizations of the corresponding reduced ordinary differential equations. The procedure leads to solutions of Bernoulli equations of non-linearity 3/2 and 2 (Riccati), respectively. Introducing the initial conditions through an imaginary phase in the traveling coordinate, we obtain all the solutions previously reported, some of them being corrected here, and showing, at the same time, the presence of interesting details of these solitary waves that have been overlooked before this investigation.

Anyons, group theory and planar physics

Relativistic and nonrelativistic anyons are described in a unified formalism by means of the coadjoint orbits of the symmetry groups in the free case as well as when there is an interaction with a constant electromagnetic field. To deal with interactions we introduce the extended Poincare and Galilei Maxwell groups.