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Asymptotic expansions of canards with poles. Application to the stationary unidimensional Schrödinger equation.

The central topic of this paper is the problem of turning points. The paradigm is the stationary unidimensional Schrodinger equation, with various 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 corresponding solution. With a change of variables, it is possible to reduce the problem of the n-th energy level to the (n − 1)-th. The first result (already proved by others methods) of the paper is an algorithm to compute the asymptotic expansion of the n-th energy level in powers of the small parameter ~. The second (new) result is an algorithm to compute an expansion of the corresponding solution. This expansion is a fraction so that the singularity is resolved. For example it is possible to determine the zero of the eigenfunctions of the Schrodinger operator up to any power of ~. The algorithms are given with Maple programs, and illustrated with a double symmetrical well as potential.