Vieira, N: Parabolic Dirac Operators and the Schrödinger Equ

In this work we study the Schrödinger problem using Clifford analysis. We use this approach to present a factorization for time dependent operators in terms of the parabolic-type Dirac operator. In the case of the heat operator we show that it is possible to construct Fischer decomposition. This decomposition can be applied in the characterization of the powers of the associated homogeneous operator. For the case of the Schrödinger operator, we will apply a regularization procedure in order to control the non-removable singularity existing in the hyperplane t=0. We will study the arising operators such as the regularized Teodorescu and Cauchy-Bitsadze operators. The properties of these operators will be used to obtain a Hodge decomposition for the regularized case and general case, in terms of the kernel of the parabolic Dirac operator. In the last chapter we study the cubic non-linear Schrödinger problem using a combination of Witt basis and finite difference approximations. We will show that it is possible to construct a discrete fundamental solution for time dependent discrete Schrödinger operator, via discrete Fourier transform and the arising symbol of the Laplace operator.