Perturbed boundary eigenvalue problem for the Schrodinger operator on an interval

A perturbed two-parameter boundary value problem is considered for a second-order differential operator on an interval with Dirichlet conditions. The perturbation is described by the potential mu(-1) V((x - x (0))E >(-1)), where 0 < E > a parts per thousand(a) 1 and mu is an arbitrary parameter such that there exists delta > 0 for which E >/mu = o(E >(delta)). It is shown that the eigenvalues of this operator converge, as E > -> 0, to the eigenvalues of the operator with no potential. Complete asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed operator are constructed.