On Expansions in the Exact and Asymptotic Eigenfunctions of the One-Dimensional Schrödinger Operator

The one-dimensional Schrödinger operator with potential growing at infinity and with a semiclassical small parameter is considered. We obtain estimates via powers of the small parameter for the remainder in the expansion of smooth sufficiently rapidly decaying functions in the exact and asymptotic eigenfunctions. For the asymptotic eigenfunctions, we use a global representation in the form of an Airy function.