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