On the reduction of the multidimensional Schrödinger equation to a first order equation and its relation to the pseudoanalytic function theory
نویسنده
چکیده
Given a particular solution of a one-dimensional stationary Schrödinger equation this equation of second order can be reduced to a first order linear ordinary differential equation. This is done with the aid of an auxiliary Riccati differential equation. In the present work we show that the same fact is true in a multidimensional situation also. For simplicity we consider the case of two or three independent variables. One particular solution of the Schrödinger equation allows us to reduce this second order equation to a linear first order quaternionic differential equation. As in one-dimensional case this is done with the aid of an auxiliary quaternionic Riccati equation. The resulting first order quaternionic equation is equivalent to the static Maxwell system and is closely related to the Dirac equation. In the case of two independent variables it is the well known Vekua equation from theory of pseudoanalytic (or generalized analytic) functions. Nevertheless we show that even in this case it is very useful to consider not complex valued functions only, solutions of the Vekua equation but complete quaternionic functions. In this way the first order quaternionic equation represents two separate Vekua equations, one of which gives us solutions of the
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