Relationship Between Bicomplex Generalized Analytic Functions and Solutions of the Complexified Schr\"odinger Equation

Abstract

Using three different representations of the bicomplex numbers $T\cong Cl_{C}(1,0) \cong Cl_{C}(0,1)$, which is a commutative ring with zero divisors defined by $T={w_0+w_1 {i_1}+w_2{i_2}+w_3 {j} | w_0,w_1,w_2,w_3 \in{R}}$ where ${i_1^{2}}=-1, {i_2^{2}}=-1, {j^{2}}=1 and {i_1}{i_2}={j}={i_2}{i_1}$, we construct three classes of bicomplex pseudoanalytic functions. In particular, we obtain some specific systems of Vekua equations of two complex variables and we established some connections between one of these systems and the classical Vekua equations. We consider also the complexification of the real stationary two-dimensional Schr{\"o}dinger equation. With the aid of any of its particular solutions, we construct a specific bicomplex Vekua equation possessing the following special property. The scalar parts of its solutions are solutions of the original complexified Schr{\"o}dinger equation and the vectorial parts are solutions of another complexified Schr{\"o}dinger equation.