ASYMPTOTIC EXPANSIONS OF CANARDS WITH POLES - APPLICATION TO THE STATIONARY UNIDIMENSIONAL SCHRODINGER-EQUATION

The central topic of this paper is the problem of turning points. The paradigm is the stationary unidimensional Schrodinger equation, with v arious potentials. The first step is to transform the linear equation of second order into a Riccati equation. The non standard analysis and the theory of canards allow to compute the first eigenvalue and the c orresponding solution. With a change of variables, it is possible to r educe the problem of the n-th energy level to the (n - 1)-th. The firs t result (already proved by others methods) of the paper is an algorit hm to compute the asymptotic expansion of the n-th energy level in pow ers of the Planck's constant. The second (new) result is an algorithm to compete an expansion of the corresponding solution. This expansion is a fraction so that the singularity is resolved. For example it is p ossible to determine the zero of the eigenfunctions of the Schrodinger operator zip to any power of the Planck's constant. The algorithms ar e implemented in Maple, and illustrated with a double symmetrical well as potential.