Multiple scales analysis of Hamiltonians with short-range potentials

The formal asymptotic expansion in epsilon of the eigenvalues and eigenfunctions of a family of one-dimensional Hamiltonians -d(2)/dx(2) + V(x) + l/epsilon W(x/epsilon) is considered in the limit epsilon -> 0, using the technique of multiple scales. For sufficiently localized short-range potentials W the 0(epsilon(0)) approximation can be found by the usual replacement of the function W by a delta-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.