The exactly solvable two-dimensional stationary Schr\"odinger operators obtaining by the nonlocal Darboux transformation

TL;DR

This paper introduces a nonlocal Darboux transformation method to derive exactly solvable two-dimensional stationary Schrödinger operators with smooth, decaying potentials, expanding the class of solvable models in quantum mechanics.

Contribution

It develops a novel nonlocal Darboux transformation framework for potential equations, enabling the construction of new exactly solvable 2D Schrödinger operators.

Findings

Derived new exactly solvable 2D Schrödinger operators

Constructed smooth potentials decaying at infinity

Provided explicit examples of solvable models

Abstract

The Fokker-Planck equation associated with the two - dimensional stationary Schr\"odinger equation has the conservation low form that yields a pair of potential equations. The special form of Darboux transformation of the potential equations system is considered. As the potential variable is a nonlocal variable for the Schr\"odinger equation that provides the nonlocal Darboux transformation for the Schr\"odinger equation. This nonlocal transformation is applied for obtaining of the exactly solvable two - dimensional stationary Schr\"odinger equations. The examples of exactly solvable two - dimensional stationary Schr\"odinger operators with smooth potentials decaying at infinity are obtained.