Multiple scales analysis of Hamiltonians with short-range potentials

The formal asymptotic expansion in ε of the eigenvalues and eigenfunctions of a family of one-dimensional Hamiltonians -d2/dx2 + V(x) + 1/εW(x/ε) is considered in the limit ε → 0, using the technique of multiple scales. For sufficiently localized short-range potentials W the O(ε0) approximation can be found by the usual replacement of the function W by a δ-type distribution. A simple analytical formula is found which expresses the first-order correction in terms of the zeroth-order wavefunction and some moments of the function W