Geodesic
Boundary-Value Problems for the Schr\"odinger Equation with Rapidly Oscillating and Delta-Liked Potentials
Matematičeskie zametki, Tome 98 (2015) no. 6, pp. 842-852

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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.
Keywords: Schrödinger equation, boundary-value problem, $1$-periodic function, homogenization method, matching method, rapidly oscillating potential, delta-liked potential. 
@article{MZM_2015_98_6_a4,
     author = {T. R. Gadylshin},
     title = {Boundary-Value {Problems} for the {Schr\"odinger} {Equation} with {Rapidly} {Oscillating} and {Delta-Liked} {Potentials}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {842--852},
     publisher = {mathdoc},
     volume = {98},
     number = {6},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2015_98_6_a4/}
}
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T. R. Gadylshin. Boundary-Value Problems for the Schr\"odinger Equation with Rapidly Oscillating and Delta-Liked Potentials. Matematičeskie zametki, Tome 98 (2015) no. 6, pp. 842-852. http://geodesic.mathdoc.fr/item/MZM_2015_98_6_a4/