Fundamentals of bicomplex pseudoanalytic function theory: Cauchy integral formulas, negative formal powers and Schr\"odinger equations with complex coefficients

TL;DR

This paper develops a comprehensive theory of bicomplex pseudoanalytic functions, including Cauchy integral formulas and formal powers, and explores their applications to Schrödinger equations with complex potentials.

Contribution

It introduces a new class of reproducing Cauchy kernels, extends classical formulas to bicomplex settings, and links these to solutions of complex Schrödinger equations.

Findings

Established a complete characterization of reproducing Cauchy kernels.

Developed an algorithm for constructing negative formal powers.

Connected bicomplex Vekua equations to Schrödinger operators and their fundamental solutions.

Abstract

The study of the Dirac system and second-order elliptic equations with complex-valued coefficients on the plane leads to bicomplex Vekua equations. To the difference of complex pseudoanalytic (generalized analytic) functions the theory of bicomplex functions has not been developed. Such basic facts as the similarity principle or the Liouville theorem in general are no longer available due to the presence of zero divisors in the algebra of bicomplex numbers. We develop a theory of bicomplex pseudoanalytic formal powers analogous to the developed by Bers and obtain Cauchy's integral formula in the bicomplex setting. In the classical complex situation this formula was obtained under the assumption that the involved Cauchy kernel is global, a restrictive condition taking into account possible applications, especially when the equation itself is not defined on the whole plane. We show that the Cauchy integral formula remains valid with a Cauchy kernel from a wider class called here the reproducing Cauchy kernels. We give a complete characterization of this class. To our best knowledge these results are new even for complex Vekua equations. We establish that reproducing Cauchy kernels can be used to obtain a full set of negative formal powers for the corresponding bicomplex Vekua equation and present an algorithm for their construction. Bicomplex Vekua equations of a special form called main Vekua equations are closely related to Schr\"odinger equations with complex-valued potentials. We use this relation to establish connections between the reproducing Cauchy kernels and the fundamental solutions for the Schr\"odinger operators which allow one to construct the Cauchy kernel when the fundamental solution is known and vice versa as well as to construct the fundamental solutions for the Darboux transformed Schr\"odinger operators.