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

TL;DR

This paper explores the application of pseudoanalytic function theory to two-dimensional supersymmetric quantum mechanics, revealing new insights into Hamiltonian factorization, ground state solutions, and the role of transmutation operators.

Contribution

It introduces a novel approach linking pseudoanalytic functions and transmutation operators to analyze supersymmetric quantum systems in two dimensions.

Findings

Ground states correspond to solutions of Vekua equations and Bers derivatives.

Darboux transformations are represented as Bers derivatives in the complex plane.

Hamiltonian components relate to the Laplacian via 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, 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.