On the reduction of the multidimensional Schroedinger equation to a first order equation and its relation to the pseudoanalytic function theory

Given a particular solution of a one-dimensional stationary Schroedinger equation (SE) this equation of second order can be reduced to a first order linear differential equation. This is done with the aid of an auxiliary Riccati equation. We show that a similar fact is true in a multidimensional situation also. We consider the case of two or three independent variables. One particular solution of (SE) 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 Riccati equation. The resulting first order quaternionic equation is equivalent to the static Maxwell system. In the case of two independent variables it is the Vekua equation from theory of generalized analytic functions. We show that even in this case it is necessary to consider not complex valued functions only, solutions of the Vekua equation but complete quaternionic functions. Then the first order quaternionic equation represents two separate Vekua equations, one of which gives us solutions of (SE) and the other can be considered as an auxiliary equation of a simpler structure. For the auxiliary equation we always have the corresponding Bers generating pair, the base of the Bers theory of pseudoanalytic functions, and what is very important, the Bers derivatives of solutions of the auxiliary equation give us solutions of the main Vekua equation and as a consequence of (SE). We obtain an analogue of the Cauchy integral theorem for solutions of (SE). For an ample class of potentials (which includes for instance all radial potentials), this new approach gives us a simple procedure allowing to obtain an infinite sequence of solutions of (SE) from one known particular solution

Quaternionic factorization of the Schroedinger operator and its applications to some first order systems of mathematical physics

We consider the following first order systems of mathematical physics. 1.The Dirac equation with scalar potential. 2.The Dirac equation with electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The system describing non-linear force free magnetic fields or Beltrami fields with nonconstant proportionality factor. 5.The Maxwell equations for slowly changing media. 6.The static Maxwell system. We show that all this variety of first order systems reduces to a single quaternionic equation the analysis of which in its turn reduces to the solution of a Schroedinger equation with biquaternionic potential. In some important situations the biquaternionic potential can be diagonalized and converted into scalar potentials

On a factorization of second order elliptic operators and applications

We show that given a nonvanishing particular solution of the equation (divpgrad+q)u=0 (1) the corresponding differential operator can be factorized into a product of two first order operators. The factorization allows us to reduce the equation (1) to a first order equation which in a two-dimensional case is the Vekua equation of a special form. Under quite general conditions on the coefficients p and q we obtain an algorithm which allows us to construct in explicit form the positive formal powers (solutions of the Vekua equation generalizing the usual powers of the variable z). This result means that under quite general conditions one can construct an infinite system of exact solutions of (1) explicitly, and moreover, at least when p and q are real valued this system will be complete in ker(divpgrad+q) in the sense that any solution of (1) in a simply connected domain can be represented as an infinite series of obtained exact solutions which converges uniformly on any compact subset of . Finally we give a similar factorization of the operator (divpgrad+q) in a multidimensional case and obtain a natural generalization of the Vekua equation which is related to second order operators in a similar way as its two-dimensional prototype does

On a complex differential Riccati equation

We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation as, e.g., the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical "one-dimensional" results we discuss new features of the considered equation like, e.g., an analogue of the Cauchy integral theorem