From Osher Doctorow
Vladislav V. Kravchenko (Depto. de Telecommunicacions, SEPI, Escuela
Superior de Ingenieria Mecanica y Electrica Instituto Politecnico
Mexico) in "On the reduction of the multidimensional Schrodinger
equation to a first order equation and its relation to the
pseudoanalytic function theory," math.AP/0408172 v1 12 Aug 2004,
started the series of 3 papers of which the other 2 were discussed here
in part 1.
It is instructive to follow his one-dimensional derivation which he
parallels in higher dimensions in his 2001 paper with two other
authors.
He starts with the one-dimensional static Schrodinger equation:
1) u" + vu = 0
which has the associated Riccati equation:
2) y' + y^2 = -v
which are related by the substitution:
3) y = -u'/u
so that the solutions of one are either the simple logarithmic
derivatives or antiderivatives of solutions of the other.
In this 2004 paper he considers for simplicity the case of 2-3
independent variables, one particular solution of Schrodinger enabling
him to reduce the second order equation to a linear first order
quaternionic differential equation with the help of an auxiliary
quaternionic Riccati equation.
Osher Doctorow
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