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

Abstract

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.

References in corpus (2)

• Two-dimensional rational solitons and their blow-up via the Moutard transformation
• 2D Schrodinger Operator, (2+1) Systems and New Reductions. The 2D Burgers Hierarchy and Inverse Problem Data

Cited by in corpus (3)

• On the nonlocal Darboux transformation for the stationary axially symmetric Schrödinger and Helmholtz equations
• The nonlocal Darboux transformation of the 2D stationary Schrödinger equation and its relation to the Moutard transformation
• The nonlocal Darboux transformation of the stationary axially symmetric Schrödinger equation and generalized Moutard transformation