Darboux Transformations for Schrodinger Equations in Two Variables

Darboux transformations in one variable form the basis for the factorization methods and have numerous applications to geometry, nonlinear equations and SUSY quantum mechanics. In spite of this wide range of applications the theory of Darboux transformations in two variables and its elegant relationship to analytic complex functions has not been recognized in the literature. To close this gap we develop in this paper the theory of Darboux transformation in the context of Schrödinger equations in two variables. This yields a constructive algorithm to determine the relationship between potential functions which are related by Darboux transformations.