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В работе рассматриваются краевые задачи на отрезке [ , ] для уравнения Шрёдингера с потенциалом в виде суммы (, −1) + −1 (−1), где (,) является 1-периодической по функцией, () есть финитная функция, 0 ∈ (,), ,-малые положительные параметры. На основе комбинации метода усреднения и метода согласования асимптотических разложений построены решения этих краевых задач с точностью до (+). Библиография: 11 названий.

Abstract. We consider boundary value problems for one-dimensional second order quasilinear equation on bounded and unbounded intervals of the real axis. The equation perturbed by the delta-shaped potential −1 (︀ −1 )︀ , where ( ) is a compactly supported function, 0 < ≪ 1. The mean value of ⟨ ⟩ can be negative, but it is assumed to be bounded from below ⟨ ⟩ − 0 . The number 0 is defined in terms of coefficients of the equation. We study the convergence rate of the solution of the perturbed problem to the solution of the limit problem 0 as the parameter tends to zero. In the case of a bounded interval , the estimate of the form | ( ) − 0 ( )| < is established. As the interval is unbounded, we prove a weaker estimate | ( ) − 0 ( )/ < 1/2 . The estimates are proved by using original cut-off functions as trial functions. For simplicity, the proof of the existence of solutions to perturbed and limiting problems are made by the method of contracting mappings. The disadvantage of this approach, as it is known, is the smallness of the nonlinearities in the equation. We consider the cases of the Dirichlet, Neumann and Robin condition.

The paper is devoted to the vibrations of a string I with a concentrated mass ε −1 Q ε −1 x and rapidly oscillating density q x, μ −1 x , where q(x, ζ ) is a 1-periodic in ζ function, Q (ξ ) is a function with compact support, the integral of which is equal to one, 0 ∈ I, μ, ε are small positive parameters, ∈ R. By combining homogenization and the method of matched asymptotic expansions, we construct solutions to the problems up to O (ε + μ).

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