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

摘要

We consider the real stationary two-dimensional Schroedinger equation. Withthe aid of any its particular solution we construct a Vekua equation possessingthe following special property. The real parts of its solutions are solutionsof the original Schroedinger equation and the imaginary parts are solutions ofan associated Schroedinger equation with a potential having the form of apotential obtained after the Darboux transformation. Using L. Bers theory ofTaylor series for pseudoanalytic functions we obtain a locally complete systemof solutions of the original Schroedinger equation which can be constructedexplicitly for an ample class of Schroedinger equations. For example it ispossible, when the potential is a function of one cartesian, spherical,parabolic or elliptic variable. We give some examples of application of theproposed procedure for obtaining a locally complete system of solutions of theSchroedinger equation. The procedure is algorithmically simple and can beimplemented with the aid of a computer system of symbolic or numericalcalculation.