Perturbed boundary eigenvalue problem for the Schrödinger operator on an interval
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- Khusnullin, I.K. Comput. Math. and Math. Phys. (2010) 50: 646. doi:10.1134/S096554251004007X
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 μ−1V((x − x0)ɛ−1), where 0 < ɛ ≪ 1 and μ is an arbitrary parameter such that there exists δ > 0 for which ɛ/μ = o(ɛδ). It is shown that the eigenvalues of this operator converge, as ɛ → 0, to the eigenvalues of the operator with no potential. Complete asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed operator are constructed.