Nelson Faustino

Fischer Decomposition for Difference Dirac Operators

Abstract.We establish the basis of a discrete function theory starting with a Fischer decomposition for difference Dirac operators. Discrete versions of homogeneous polynomials, Euler and Gamma operators are obtained. As a consequence we obtain a Fischer decomposition for the discrete Laplacian.

Difference potentials for the Navier–Stokes equations in unbounded domains

We develop a numerical method for the Navier–Stokes equations over unbounded domains. From the analytic methods used to show existence and uniqueness, we obtain their discrete counterparts which allows us to establish a problem-adapted numerical solver based on finite differences for functions with low regularity.

(DISCRETE) ALMANSI TYPE DECOMPOSITIONS: AN UMBRAL CALCULUS FRAMEWORK BASED ON osp(1|2) SYMMETRIES

We introduce the umbral calculus formalism for hypercomplex variables starting from the fact that the algebra of multi- variate polynomials IRŒxshall be described in terms of the generators of the Weyl-Heisenberg algebra. The extension of IRŒxto the algebra of Clifford-valued polynomials P gives rise to an algebra of Clifford-valued operators whose canonical generators are isomorphic to the orthosymplectic Lie algebra osp.1j2/. This extension provides an effective framework in continuity and discreteness that allow us to establish an alternative formulation of Almansi decomposition in Clifford analysis obtained by Ryan (Zeitschrift fur Analysis und ihre Anwendun- gen1990) and Malonek & Ren (MathematicalMethodsintheAppliedSciences2002;2007) that corresponds to a meaningful generalization of Fischer decomposition for the subspaces ker.D 0 / k . We will discuss afterwards how the symmetries of sl2.IR/ (even part of osp.1j2/) are ubiquitous on the recent approach of RENDER (Duke Mathematical Journal 2008) show- ingthattheycanbeinterpretedintermsofthemethodofseparationofvariablesfortheHamiltonianoperatorinquantum mechanics. Copyright

Fock spaces, Landau operators and the time-harmonic Maxwell equations

We investigate the representations of the solutions to Maxwells equations based on the combination of hypercomplex function-theoretical methods with quantum mechanical methods. Our approach provides us with a characterization for the solutions to the time-harmonic Maxwell system in terms of series expansions involving spherical harmonics resp. spherical monogenics. Also, a thorough investigation for the series representation of the solutions in terms of eigenfunctions of Landau operators that encode n-dimensional spinless electrons is given. This new insight should lead to important investigations in the study of regularity and hypo-ellipticity of the solutions to Schrodinger equations with natural applications in relativistic quantum mechanics concerning massive spinor fields.

Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature ( 0 , n ) the umbral calculus framework with Lie-algebraic symmetries. The exponential generating function (EGF) carrying the continuum Dirac operator D = ? j = 1 n e j ? x j together with the Lie-algebraic representation of raising and lowering operators acting on the lattice h Z n is used to derive the corresponding hypercomplex polynomials of discrete variable as Appell sets with membership on the space Clifford-vector-valued polynomials. Some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the Poisson-Charlier polynomials are introduced. Certain applications from the view of interpolation theory and integral transforms are also discussed.

Numerical Clifford analysis for nonlinear Schrödinger problem

The aim of this work is to study the numerical solution of the nonlinear Schrodinger problem using a combination between Witt basis and finite difference approximations. We construct a discrete fundamental solution for the nonstationary Schrodinger operator and we show the convergence of the numerical scheme. Numerical examples are given at the end of the article.

Special Functions of Hypercomplex Variable on the Lattice Based on SU(1,1)

Based on the representation of a set of canonical operators on the lattice hZ n , which are Clifford-vector-valued, we will introduce new families of special functions of hy- percomplex variable possessing su(1; 1) symmetries. The Fourier decomposition of the space of Clifford-vector-valued polynomials with respect to the SO(n) su(1; 1)-module gives rise to the construction of new families of polynomial sequences as eigenfunctions of a coupled system involving forward/backward discretizations E h of the Euler operator E = n P j=1 x j@xj . Moreover, the interpretation of the one-parameter representation Eh(t) = exp(tE h tE + h ) of

Solutions for the Klein–Gordon and Dirac Equations on the Lattice Based on Chebyshev Polynomials

The main goal of this paper is to adopt a multivector calculus scheme to study finite difference discretizations of Klein–Gordon and Dirac equations for which Chebyshev polynomials of the first kind may be used to represent a set of solutions. The development of a well-adapted discrete Clifford calculus framework based on spinor fields allows us to represent, using solely projection based arguments, the solutions for the discretized Dirac equations from the knowledge of the solutions of the discretized Klein–Gordon equation. Implications of those findings on the interpretation of the lattice fermion doubling problem is briefly discussed.

Numerical Clifford Analysis for the Non‐stationary Schrödinger Equation

We construct a discrete fundamental solution for the parabolic Dirac operator which factorizes the non‐stationary Schrodinger operator. With such fundamental solution we construct a discrete counterpart for the Teodorescu and Cauchy‐Bitsadze operators and the Bergman projectors. We finalize this paper with convergence results regarding the operators and a concrete numerical example.