Introduction 1 Preliminaries 1.1 Clifford Algebras 1.1.1 Real Clifford algebras 1.1.2 Complex Clifford algebra 1.1.3 Function spaces 1.2 Clifford analysis 1.2.1 Teodorescu operator 1.2.2 Clifford analysis using the Teodorescu operator 2 Continuous calculus operator 2.1 Time-dependent operators 2.1.1 Factorization of time-dependent operators 2.1.2 Fischer decomposition for the homogeneous operator D 2.1.3 Powers of D 2.2 Operator calculus for the SchrSdinger operator 2.2.1 Regularization of the fundamental solution 2.2.2 Regularized Teodorescu and Cauchy-Bitsadze operators 2.2.3 Hypoelliptic analysis 2.2.4 Lp-decomposition 3 The non-linear Schrodinger problem 3.1 Resolution of the NLS problem via an iterative method 3.1.1 Existence and uniqueness of solution 3.1.2 Convergent iterative method 3.2 Discrete fundamental solution for time-evolution problems 3.2.1 Quaternionic matrix representation of the Witt basis 3.2.2 Finite differences and time evolution operators 3.2.3 Discrete symbol of the Laplace operator 3.2.4 Discrete fundamental solutions 3.3 Discrete operator calculus 3.3.1 Behavior of the discrete fundamental solution 3.3.2 Discrete operators 3.3.3 Numerical examples Conclusion Appendix A Hypoelliptic Theory A.1 Definition and main properties A.2 Sufficient conditions for hypoenipticity Bibliography Index ��݋��ӛ