A perturbed boundary eigenvalue problem for the Schrödinger 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 $\mu^{-1}V((x-x_0)\varepsilon^{-1})$, where $0<\varepsilon\ll1$ and $\mu$ is an arbitrary parameter such that there exists $\delta>0$ for which $\varepsilon/\mu=o(\varepsilon^\delta)$. It is shown that the eigenvalues of this operator converge, as $\varepsilon\to0$, to the eigenvalues of the operator with no potential. Complete asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed operator are constructed.