On a relation of pseudoanalytic function theory to the two-dimensional stationary Schroedinger equation and Taylor series in formal powers for its solutions

TL;DR

This paper links pseudoanalytic function theory to the 2D stationary Schrödinger equation, constructing explicit solution systems via Taylor series in formal powers, applicable to various potential functions.

Contribution

It introduces a method to generate complete solution systems for the Schrödinger equation using pseudoanalytic functions and Taylor series, applicable to diverse potentials.

Findings

Constructed solutions include real parts solving the original Schrödinger equation.

Solutions can be explicitly constructed for potentials depending on common variables.

The method is algorithmically simple and suitable for computer implementation.

Abstract

We consider the real stationary two-dimensional Schroedinger equation. With the aid of any its particular solution we construct a Vekua equation possessing the following special property. The real parts of its solutions are solutions of the original Schroedinger equation and the imaginary parts are solutions of an associated Schroedinger equation with a potential having the form of a potential obtained after the Darboux transformation. Using L. Bers theory of Taylor series for pseudoanalytic functions we obtain a locally complete system of solutions of the original Schroedinger equation which can be constructed explicitly for an ample class of Schroedinger equations. For example it is possible, when the potential is a function of one cartesian, spherical, parabolic or elliptic variable. We give some examples of application of the proposed procedure for obtaining a locally complete system of solutions of the Schroedinger equation. The procedure is algorithmically simple and can be implemented with the aid of a computer system of symbolic or numerical calculation.