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

Abstract

Pseudoanalytic function theory is considered to study a two-dimensional supersymmetric quantum mechanics system. Hamiltonian components of the superHamiltonian are factorized in terms of one Vekua and one Bers derivative operators. We show that imaginary and real solutions of a Vekua equation and its Bers derivative are ground state solutions for the superHamiltonian. The two-dimensional Darboux and pseudo-Darboux transformations correspond to Bers derivatives in the complex plane. Results on the completeness of the ground states are obtained. Finally, the superpotential is studied in the separable case in terms of transmutation operators. We show how Hamiltonian components of the superHamiltonian are related to the Laplacian operator using these transmutation operators.