Introduction

The stationary Schrödinger equation is considered under the axially symmetric approach. A nonlocal Darboux transformation of the equation is evaluated and it is evidenced that a particular instance of the equation paves the way for the generalization of the Moutard transformation. By applying the generalized Moutard transformation, the acquisition of new formulas is possible. Additionally, the application of the transformation enables the derivation of two-dimensional potentials as well as precise solutions for the axially symmetric stationary Schrödinger equation.

Nonlocal Darboux Transformation

The Darboux transformation is a helpful tool for the one-dimensional Schrödinger equation. The Moutard transformation is a two-dimensional generalization of the Darboux transformation employed for the Schrödinger equation in cartesian coordinates.

In this regard, papers have considered the nonlocal Darboux transformation of the two-dimensional stationary Schrödinger equation in cartesian coordinates and established its relation to the Moutard transformation. This paper takes this concept and applies it to the stationary Schrödinger equation in cylindrical coordinates, following the approach of the aforementioned references.

The Nonlocal Darboux Transformation and Generalization of the Moutard Transformation

The equations for the coefficients and the other parameters can be derived after considering both the linear operator and the Darboux transformation for the equations system. While solving these equations, a special situation in the form of a formula for Darboux transformation arises. This formula leads to another useful expression - the generalization of the Moutard transformation for the potential of the Schrödinger equation.

In the end, these formulas prove to be the generalization of the Moutard transformation for the solution of the Schrödinger equation. Thus, it is evident that a specific case of the nonlocal Darboux transformation for the stationary axially symmetric Schrödinger equation in fact provides a generalized form of the Moutard transformation.

Application of the Generalized Moutard Transformation

The first practical application of the generalized Moutard transformation is to consider the scenario where u equals to 0. Applying the transformation provides us with a potential formula which has singularities. The two-dimensional Schrödinger equation in cartesian coordinates can be effectively dealt with by using the Moutard transformation twice to get nonsingular potentials.

Adopting this approach, we apply the generalized Moutard transformation for the second time on our example and get nonsingular potential in the case when C1 is greater than zero. To provide solutions for the stationary axially symmetric Schrödinger equation, the generalized Moutard transformation proves to be an effective tool.