Computational Mathematics and Mathematical Physics

, Volume 50, Issue 4, pp 646-664

Perturbed boundary eigenvalue problem for the Schrödinger operator on an interval

  • I. Kh. KhusnullinAffiliated withBashkir State Pedagogical University Email author 

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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−1 V((xx 0−1), where 0 < ɛ ≪ 1 and μ is an arbitrary parameter such that there exists δ > 0 for which ɛ/μ = oδ). 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.

Key words

second-order differential operator singular perturbation eigenvalue asymptotics