Boundary-value problems for the Schröinger equation with rapidly oscillating and delta-liked potentials

Abstract

This paper deals with boundary-value problems on the closed interval [a, b] for the Schrödinger equation with potential of the form q(x, μ ⁻¹ x) + ε ⁻¹ Q(ε ⁻¹ x), where q(x, ζ) is a 1-periodic (in ζ) function, Q(ξ) is a compactly supported function, 0 ∈ (a, b), and μ, ε are small positive parameters. The solutions of these boundary-value problemsup to O(ε +μ) are constructed by combining the homogenization method and the method of matching asymptotic expansions.