On a relation of bicomplex pseudoanalytic function theory to the complexified stationary Schrödinger equation

Using three different representations of the bicomplex numbers T ∼= ClC(1, 0) ∼= ClC(0, 1), which is a commutative ring with zero divisors defined by T = {w0 + w1i1 + w2i2 + w3j | w0, w1, w2, w3 ∈ R} where i1 = −1, i 2 2 = −1, j 2 = 1 and i1i2 = j = i2i1, 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 Schrödinger 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 Schrödinger equation and the vectorial parts are solutions of another complexified Schrödinger equation.