On a relation of pseudoanalytic function theory to the two-dimensional stationary Schrodinger equation and Taylor series in formal powers for its solutions

摘要

We consider the real stationary two-dimensional Schrodinger equation. With the aid of any of its particular solutions, we construct a Vekua equation possessing the following special property. The real parts of its solutions are solutions of the original Schrodinger equation and the imaginary parts are solutions of an associated Schrodinger equation with a potential having the form of a potential obtained after the Darboux transformation. Using Bers' theory of Taylor series for pseudoanalytic functions, we obtain a locally complete system of solutions of the original Schrodinger equation which can be constructed explicitly for an ample class of Schrodinger equations. For example it is possible when the potential is a function of one-Cartesian, spherical, parabolic or elliptic variable. We give some examples of application of the proposed procedure for obtaining a locally complete system of solutions of the Schrodinger equation. The procedure is algorithmically simple and can be implemented with the aid of a computer system of symbolic or numerical calculation.