WKB-Langer Asymptotic L Approximation of Eigenfunctions and Their Derivatives

In this paper we study the WKB-Langer asymptotic expansion of the eigenfunctions of a Schrödinger operator H = − 12 2 ∂2 ∂x2 + V (x). Then applying these asymptotic formulae we prove that the exact L eigenfunction ΨE(N, ) and its derivative ΨE(N, ) of the Schrödinger operator with a well-shaped analytic potential are approximated up to arbitrary order m by the semi-classical WKB-Langer approximate eigenfunction ΨEm(N, ),m and its derivative Ψ ′ Em(N, ),m respectively in L, i.e. ||ΨE(N, ) − ΨEm(N, ),m||L2 = O( ), || ΨE(N, ) − ΨEm(N, ),m||L2 = O( ) uniformly for any N . Here Em(N, ) approximates E(N, ) up to m-th order (in ) and satisfies m-th order quantization condition.