Boundary-Value Problems for the Schrödinger Equation with Rapidly Oscillating and Delta-Liked Potentials

This paper deals with boundary-value problems on the closed interval $[a,b]$ for the Schrödinger equation with potential of the form $q(x,\mu^{-1}x)+\varepsilon^{-1}Q(\varepsilon^{-1}x)$, where $q(x,\zeta)$ is a $1$-periodic (in $\zeta$) function, $Q(\xi)$ is a compactly supported function, $0\in(a,b)$, and $\mu,\varepsilon$ are small positive parameters. The solutions of these boundary-value problems up to $O(\varepsilon+\mu)$ are constructed by combining the homogenization method and the method of matching asymptotic expansions.