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On explicitly solvable Vekua equations and explicit solution of the stationary Schrödinger equation and of the equation div($\sigma \nabla u\right)=0$. (English) Zbl 1122.30029
There is the well-known relation in complex analysis-between holomorphic functions and pairs of conjugate harmonic functions. The authors are interested to describe generalizations of this relation, namely between solutions from special classes of Vekua’s equation and pairs of associated stationary Schrödinger equations in $2-d$ with potentials. If $\left({w}_{1},{w}_{2}\right)$ is such a pair of solutions of associated Schrödinger equations, then $w={w}_{1}+i{w}_{2}$ represents a solution of a special Vekua equation and vice versa. It is shown that such a relation can be generalized to solutions of associated pairs of equations $\nabla ·\left({a}_{1}\left(x,y\right)\nabla {w}_{1}\right)=0$ and $\nabla ·\left({a}_{2}\left(x,y\right)\nabla {w}_{2}\right)=0$ in the above described sense. If ${w}_{1}$ is given, then the construction of ${w}_{2}$ is proposed. Finally, the authors discuss the question if from the explicit solvability of one Vekua equation the explicit solvability of other ones can be concluded.
MSC:
 30G20 Generalizations of analytic functions of Bers or Vekua type 35J10 Schrödinger operator