ASYMPTOTIC (SHORT-WAVE) EQUIVALENCE OF ONE-DIMENSIONAL SCHRODINGER-EQUATIONS BY FORMAL CANONICAL-TRANSFORMATIONS AND ITS GENERALIZATIONS

We define the asymptotic equivalence of systems of two ordinary first-order differential equations with an arbitrary finite number and multiplicities of turning points. The theory is exemplified by the short-wave (semiclassical) equivalence of time-independent one-dimensional Schrodinger equations. We describe the set of all transformation matrices realizing the equivalence; the use of their determinant properties simplifies the calculations needed for applications. For the particular case of Schrodinger equations the transformation matrix can be chosen to be canonical.