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The exactly solvable two-dimensional stationary Schrödinger operators obtaining by the nonlocal Darboux transformation

The Fokker-Planck equation associated with the two - dimensional stationary Schrödinger 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 Schrödinger equation that provides the nonlocal Darboux transformation for the Schrödinger equation. This nonlocal transformation is applied for obtaining of the exactly solvable two - dimensional stationary Schrödinger equations. The examples of exactly solvable two - dimensional stationary Schrödinger operators with smooth potentials decaying at infinity are obtained.

preprint2012arXivOpen access
Andrey Kudryavtsev
math.APmath-phmath.MPquant-ph
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Andrey Kudryavtsev

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