On the Klein-Gordon equation and hyperbolic pseudoanalytic function theory

Abstract

Elliptic pseudoanalytic function theory was considered independently by Bers and Vekua decades ago. In this paper we develop a hyperbolic analogue of pseudoanalytic function theory using the algebra of hyperbolic numbers. We consider the Klein-Gordon equation with a potential. With the aid of one particular solution we factorize the Klein-Gordon operator in terms of two Vekua-type operators. We show that real parts of the solutions of one of these Vekua-type operators are solutions of the considered Klein-Gordon equation. Using hyperbolic pseudoanalytic function theory, we then obtain explicit construction of infinite systems of solutions of the Klein-Gordon equation with potential. Finally, we give some examples of application of the proposed procedure.

References in corpus (2)

• On a relation of pseudoanalytic function theory to the two-dimensional stationary Schroedinger equation and Taylor series in formal powers for its solutions
• On a factorization of second order elliptic operators and applications

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• Zakharov-Shabat system and hyperbolic pseudoanalytic function theory
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• Introduction to $\mathcal{A}$-Calculus
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• Relationship Between Bicomplex Generalized Analytic Functions and Solutions of the Complexified Schrödinger Equation