Transmutations, L-bases and complete families of solutions of the stationary Schr\"odinger equation in the plane

TL;DR

This paper develops a transmutation operator framework for the stationary Schrödinger equation in the plane, establishing the completeness of an infinite solution system using bicomplex pseudoanalytic functions, enabling advanced boundary and eigenvalue problem solutions.

Contribution

It introduces a new transmutation operator linking powers of the independent variable to an L-basis, proving the completeness of a solution system for the Schrödinger equation, and advances pseudoanalytic function theory applications.

Findings

Established a precise form of the transmutation operator.

Proved the completeness of an infinite solution system.

Enabled new methods for boundary and eigenvalue problems.

Abstract

An L-basis associated to a linear second-order ordinary differential operator L is an infinite sequence of functions {\phi_k}_{k=0}^{\infty} such that L\phi_k=0 for k=0,1, L\phi_k=k(k-1)\phi_{k-2}, for k=2,3,... and all \phi_k satisfy certain prescribed initial conditions. We study the transmutation operators related to L in terms of the transformation of powers of the independent variable {(x-x_{0})^k}_{k=0}^{\infty} to the elements of the L-basis and establish a precise form of the transmutation operator realizing this transformation. We use this transmutation operator to establish a completeness of an infinite system of solutions of the stationary Schr\"odinger equation from a certain class. The system of solutions is obtained as an application of the theory of bicomplex pseudoanalytic functions and its completeness was a long sought result. Its use for constructing reproducing kernels and solving boundary and eigenvalue problems has been considered even without the required completeness justification. The obtained result on the completeness opens the way for further development and application of the tools of pseudoanalytic function theory.