Geodesic
On approximation of the “Membrane” Schrödinger operator by the “Crystal” operator
Matematičeskie zametki, Tome 62 (1997) no. 5, pp. 773-781 
Yu. P. Chuburin. On approximation of the “Membrane” Schrödinger operator by the “Crystal” operator. Matematičeskie zametki, Tome 62 (1997) no. 5, pp. 773-781. http://geodesic.mathdoc.fr/item/MZM_1997_62_5_a13/
@article{MZM_1997_62_5_a13,
     author = {Yu. P. Chuburin},
     title = {On approximation of the {{\textquotedblleft}Membrane{\textquotedblright}} {Schr\"odinger} operator by the {{\textquotedblleft}Crystal{\textquotedblright}} operator},
     journal = {Matemati\v{c}eskie zametki},
     pages = {773--781},
     year = {1997},
     volume = {62},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_62_5_a13/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $V(x)$, $x=(s_1,x_2,x_3)$, be a potential periodic in $x_1,x_2$ and exponentially decreasing as $|x_3|\to\infty$, and let $V_N(x)$ be the sum of shifts $V\bigl(x-(0,0,Nn_3)\bigr)$ over all integer $n_3$. We prove that the spectrum and eigenfunctions (not necessarily in the class $L^2$) of the Schrödinger operator with potential $V_N$, considered in a box, approximate the spectrum and eigenfunctions of the operator with potential $V$ and, for the negative part of the spectrum, the approximation converges exponentially in $N\to\infty$.

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