Geodesic
Lower part of the~spectrum for the~two-dimensional Schr\"odinger operator periodic in one variable and application to quantum dimers
Teoretičeskaâ i matematičeskaâ fizika, Tome 188 (2016) no. 2, pp. 288-317

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We study the semiclassical asymptotic approximation of the spectrum of the two-dimensional Schrödinger operator with a potential periodic in $x$ and increasing at infinity in $y$. We show that the lower part of the spectrum has a band structure (where bands can overlap) and calculate their widths and dispersion relations between energy and quasimomenta. The key role in the obtained asymptotic approximation is played by librations, i.e., unstable periodic trajectories of the Hamiltonian system with an inverted potential. We also present an effective numerical algorithm for computing the widths of bands and discuss applications to quantum dimers.
Keywords: periodic Schrödinger operator, spectrum, tunneling effect, spectral band
Mots-clés : dispersion relation. 
@article{TMF_2016_188_2_a5,
     author = {A. Yu. Anikin and S. Yu. Dobrokhotov and M. I. Katsnel'son},
     title = {Lower part of the~spectrum for the~two-dimensional {Schr\"odinger} operator periodic in one variable and application to quantum dimers},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {288--317},
     publisher = {mathdoc},
     volume = {188},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2016_188_2_a5/}
}
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A. Yu. Anikin; S. Yu. Dobrokhotov; M. I. Katsnel'son. Lower part of the~spectrum for the~two-dimensional Schr\"odinger operator periodic in one variable and application to quantum dimers. Teoretičeskaâ i matematičeskaâ fizika, Tome 188 (2016) no. 2, pp. 288-317. http://geodesic.mathdoc.fr/item/TMF_2016_188_2_a5/