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arXiv · 1211.5685

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

Andrey Kudryavtsev
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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.

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Andrey Kudryavtsev. 2012-11-24. The exactly solvable two-dimensional stationary Schr\"odinger operators obtaining by the nonlocal Darboux transformation. https://doi.org/10.1016/j.physleta.2013.07.036

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