Geodesic
Schr\"odinger operator eigenvalue (resonance) on a zone boundary
Teoretičeskaâ i matematičeskaâ fizika, Tome 126 (2001) no. 2, pp. 196-205

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For a Schrödinger operator with a periodic potential perturbed by a function periodic with respect to two variables and tending to zero with respect to the third variable, conditions are found under which a level (eigenvalue or resonance) falls on a zone boundary. The passage of the level through the boundary under variation of the perturbation magnitude is discussed. 
@article{TMF_2001_126_2_a1,
     author = {Yu. P. Chuburin},
     title = {Schr\"odinger operator eigenvalue (resonance) on a zone boundary},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {196--205},
     publisher = {mathdoc},
     volume = {126},
     number = {2},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2001_126_2_a1/}
}
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Yu. P. Chuburin. Schr\"odinger operator eigenvalue (resonance) on a zone boundary. Teoretičeskaâ i matematičeskaâ fizika, Tome 126 (2001) no. 2, pp. 196-205. http://geodesic.mathdoc.fr/item/TMF_2001_126_2_a1/