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On a relation of pseudoanalytic function theory to the two-dimensional stationary Schrödinger equation and Taylor series in formal powers for its solutions. (English) Zbl 1067.81032
Summary: We consider the real stationary two-dimensional Schrödinger 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 Schrödinger equation and the imaginary parts are solutions of an associated Schrödinger 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 Schrödinger equation which can be constructed explicitly for an ample class of Schrödinger 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 Schrödinger equation. The procedure is algorithmically simple and can be implemented with the aid of a computer system of symbolic or numerical calculation.
81Q10Selfadjoint operator theory in quantum theory, including spectral analysis
81Q15Perturbation theories for operators and differential equations
81Q05Closed and approximate solutions to quantum-mechanical equations
35J10Schrödinger operator