Fundamentals of Bicomplex Pseudoanalytic Function Theory: Cauchy Integral Formulas, Negative Formal Powers and Schrödinger Equations with Complex Coefficients

Abstract

The study of the Dirac system and second-order elliptic equations with complexvalued coefficients on the plane naturally leads to bicomplex Vekua-type equations [8], [14], [6]. To the difference of complex pseudoanalytic (or generalized analytic) functions [3], [25] the theory of bicomplex pseudoanalytic functions has not been developed. Such basic facts as, e.g., 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.In the present work we develop a theory of bicomplex pseudoanalytic formal powers analogous to the developed by L. Bers [3] and especially that of negative formal powers. Combining the approaches of L. Bers and I. N. Vekua with some additional ideas we obtain the Cauchy 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 very restrictive condition taking into account possible practical applications, especially when the equation itself is not defined on the whole plane. We show that the Cauchy integral formula remains valid with the 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 which allows one their construction.Bicomplex Vekua equations of a special form called main Vekua equations are closely related to stationary Schrödinger equations with complex-valued potentials. We use this relation to establish useful connections between the reproducing Cauchy kernels and the fundamental solutions for the Schrödinger operators which allow one to construct the Cauchy kernel when the fundamental solution is known and vice versa. Moreover, using these results we construct the fundamental solutions for the Darboux transformed Schrödinger operators.

The Bergman kernel for the Vekua equation

“…Let Q Ω be the operator defined on L 2 (Ω) by the right-hand side of this equality. According to (6), if ϕ ∈ L 2 (Ω) there exist W ∈ H 2 (Ω) and ψ ∈ H 2 (Ω) ⊥ such that ϕ = W + ψ. Since ψ is orthogonal to both K(ζ, z) and L(ζ, z) (both kernels belong to H 2 (Ω)) then Q Ω ψ = 0 and hence…”

“…Remark 18 An analogue of the Runge theorem from complex analysis is available for the Vekua equation, replacing the usual powers of complex analysis by a special countable system of solutions of the Vekua equation called formal powers [5]. There are general conditions under which the system of formal powers can be constructed by a simple algorithm [5], [11], [7], [6]. The previous proposition is obviously related to the Runge theorem due to uniform convergence on compact sets.…”

On Sturm–Liouville equations with several spectral parameters

“…Since its appearance in 2008, consequences of this SPPS (spectral parameter power series) representation have been investigated in many directions. These include completeness properties of the "formal powers" used to define the coefficients of the power series [21,22]; relationship to transmutation operators, Darboux and other transformations, and Goursat problems [17,25,27,28]; extension to other number systems (quaternions, etc) [8,9,27] and equations of higher order [15]; relaxation of regularity conditions on the coefficients of the differential equation [5,11]. Further, there have appeared numerous applications to problems in physics and engineering [10,16,18,19,29] as well as in complex analysis [6,24].…”

On two-dimensional supersymmetric quantum mechanics, pseudoanalytic functions and transmutation operators

“…Moreover, we have shown how transmutation operators are related to Vekua, Bers derivative, Bers integral operators and hamiltonian components of the superhamiltonian. Our approach can also be applied to the two-dimensional SUSY QM system considered in this work with the superpotential χ being a complex function, though in this case complex numbers become insufficient, and one should consider the bicomplex pseudoanalytic function theory [8].…”