Factorization of the Non-Stationary Schrodinger Operator

We consider a factorization of the non-stationary Schrodinger operator based on the parabolic Dirac operator introduced by Cerejeiras/ Kahler/ Sommen. Based on the fundamental solution for the parabolic Dirac operators, we shall construct appropriated Teodorescu and Cauchy-Bitsadze operators. Afterwards we will describe how to solve the nonlinear Schrodinger equation using Banach fixed point theorem.Comment: Accepted for publication in Advances in Applied Clifford Algebra

Ternary Clifford Algebras

Ternary Clifford algebras are an essential ingredient in a cubic factorization of the Laplacian and using a ternary Clifford analysis build on such spaces one obtains a Dirac-type operator D such that D3 = Δ. This paper is a continuation of the work of the authors in describing properties of generalized ternary Clifford algebras. Here we explore a blade decomposition and symmetries of these algebras

Some applications of parabolic Dirac Operators to the instationary Navier-Stokes problem on conformally flat cylinders and tori in R^3

In this paper we give a survey on how to apply recent techniques of Clifford analysis over conformally flat manifolds to deal with instationary flow problems on cylinders and tori. Solutions are represented in terms of integral operators involving explicit expressions for the Cauchy kernel that are associated to the parabolic Dirac operators acting on spinor sections of these manifolds.publishe

Maximum principle for the regularized Schrödinger operator

In this paper we present analogues of the maximum principle and of some parabolic inequalities for the regularized time-dependent Schrödinger operator on open manifolds using Günter derivatives. Moreover, we study the uniqueness of bounded solutions for the regularized Schrödinger-Günter problem and obtain the corresponding fundamental solution. Furthermore, we present a regularized Schrödinger kernel and prove some convergence results. Finally, we present an explicit construction for the fundamental solution to the Schrödinger-Günter problem on a class of conformally flat cylinders and tori

Global Operator Calculus on Spin Groups

Acknowledgements The work of P. Cerejeiras, M. Ferreira, and U. Kähler was supported by Portuguese funds through CIDMA-Center for Research and Development in Mathematics and Applications, and FCT– “Fundação para a Ciência e a Tecnologia”, within project UIDB/04106/2020 and UIDP/04106/2020. The present paper was supported by the project “Global operator calculi on compact and non-compact Lie groups”, Ações Integradas Luso-Alemãs – Acção No. A-42/16. Funding Open access funding provided by FCT|FCCN (b-on).n this paper, we use the representation theory of the group Spin(m) to develop aspects of the global symbolic calculus of pseudo-differential operators on Spin(3) and Spin(4) in the sense of Ruzhansky–Turunen–Wirth. A detailed study of Spin(3) and Spin(4)-representations is made including recurrence relations and natural differential operators acting on matrix coefficients. We establish the calculus of left-invariant differential operators and of difference operators on the group Spin(4) and apply this to give criteria for the subellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of first and second order globally hypoelliptic differential operators are given, including some that are locally neither invertible nor hypoelliptic. The paper presents a particular case study for higher dimensional spin groups.info:eu-repo/semantics/publishedVersio

Bicomplex signals with sparsity constraints

In this paper, we aim to prove the possibility to reconstruct a bicomplex sparse signal, with high probability, from a reduced number of bicomplex random samples. Due to the idempotent representation of the bicomplex algebra, this case is similar to the case of the standard Fourier basis, thus allowing us to adapt in a rather easy way the arguments from the recent works of Rauhut and Candés et al.publishe

A wavelet based numerical method for nonlinear partial differential equations

The purpose of this paper is to present a wavelet–Galerkin scheme for solving nonlinear elliptic partial differential equations. We select as trial spaces a nested sequence of spaces from an appropriate biorthogonal multiscale analysis. This gives rise to a nonlinear discretized system. To overcome the problems of nonlinearity, we apply the machinery of interpolating wavelets to obtain knot oriented quadrature rules. Finally, Newton’s method is applied to approximate the solution in the given ansatz space. The results of some numerical experiments with different biorthogonal systems, confirming the applicability of our scheme, are presented.Instituto de Cooperação Científica e Tecnológica Internacional - Acções Integradas Luso-Alemãs (DAAD/ICCTI) - Projecto DAAD/ICCTI nº 01141

Fischer decomposition in generalized fractional ternary Clifford analysis

This paper describes the generalized fractional Clifford analysis in the ternary setting. We will give a complete algebraic and analytic description of the spaces of monogenic functions in this sense, their analogous Fischer decomposition, concluding with a description of the basis of the space of fractional homogeneous monogenic polynomials that arise in this case and an explicit algorithm for the construction of this basis