Asymptotic (semiclassical) equivalence of Schroedinger equations with singular potentials and for related systems of two first order equations

The asymptotic equivalence of systems of two ordinary first-order linear differential equations with complex independent variable and a small parameter at the derivatives is analyzed in the case of arbitrary numbers and multiplicities of turning points and singular points. The set of all the transformation matrices realizing the equivalence is described and a recursive procedure for constructing these matrices is developed. By persistently using the determinant properties of the transformation matrices, the number of integration operations at each step of this procedure is halved compared with the algorithms known before. The theory is specialized to the case of time-independent one-dimensional Schrodinger equations with singular potentials, Some generalizations to multichannel Schrodinger equations are also presented.