Geodesic
Homogenization of a~multidimensional periodic elliptic operator in a~neighbourhood of the edge of the internal gap
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 36, Tome 318 (2004), pp. 60-74

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The homogenization procedure for a multidimensional periodic Schrödinger operator near the edge of an internal gap is discussed. Approximation for the resolvent in the small period limit, with respect to the operator norm in $L_2(\mathbb{R}^d)$, is obtained. This approximation contains oscillations, but in a simpler form than the resolvent of the initial operator. 
@article{ZNSL_2004_318_a3,
     author = {M. Sh. Birman and T. A. Suslina},
     title = {Homogenization of a~multidimensional periodic elliptic operator in a~neighbourhood of the edge of the internal gap},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {60--74},
     publisher = {mathdoc},
     volume = {318},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_318_a3/}
}
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M. Sh. Birman; T. A. Suslina. Homogenization of a~multidimensional periodic elliptic operator in a~neighbourhood of the edge of the internal gap. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 36, Tome 318 (2004), pp. 60-74. http://geodesic.mathdoc.fr/item/ZNSL_2004_318_a3/