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

TL;DR

This paper explores the relationship between bicomplex generalized analytic functions and solutions to the complexified Schrödinger equation, introducing new classes of functions and connecting them to classical equations.

Contribution

It constructs three classes of bicomplex pseudoanalytic functions and links them to solutions of the complexified Schrödinger equation using Vekua equations.

Findings

Established connections between bicomplex functions and Schrödinger solutions.

Constructed specific bicomplex Vekua equations with special properties.

Demonstrated scalar parts solve the original Schrödinger 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.