Geodesic
Schr\"odinger operator in a~periodic waveguide on the plane and quasi-conformal mappings
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 33, Tome 295 (2003), pp. 204-243

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A magneric Schrödinger operator with variable metric in a two-dimensional simply connected periodic waveguide is considered. All coefficients are assumed to be periodic along the waveguide. We investigate Dirichlet and Neumann boundary problems as well as boundary problem of the third type. Under wide conditions on the boundary of the waveguide providing band structure of the spectrum, we prove the absolute continuity of the spectrum. 
@article{ZNSL_2003_295_a8,
     author = {R. G. Shterenberg},
     title = {Schr\"odinger operator in a~periodic waveguide on the plane and quasi-conformal mappings},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {204--243},
     publisher = {mathdoc},
     volume = {295},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a8/}
}
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R. G. Shterenberg. Schr\"odinger operator in a~periodic waveguide on the plane and quasi-conformal mappings. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 33, Tome 295 (2003), pp. 204-243. http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a8/