On a complete analysis of high-energy scattering matrix asymptotics for one dimensional Schrödinger operators with integrable potentials

For the general one dimensional Schrödinger operator $-\frac{d^{2}}{dx^{2}}+q\left(x\right)$ with real $q\in L_{1} \left(\mathbb{R}\right)$ we present a complete streamlined treatment of large spectral parameter power asymptotics of Jost solutions and the scattering matrix. We find simple necessary and sufficient conditions relating the number of exact terms in the asymptotics with the smoothness of $q$. These conditions are expressed in terms of the Fourier transform of some functions related to $q$. In particular, under the usual conditions $q^{\left( N\right) }\in L_{1}\left(\mathbb{R}\right), N\in\mathbb{N} _{0},$ we derive up to two extra terms in the asymptotic expansion of the Jost solution and for the transmission coefficient we derive twice as many terms. Our main results are complete.