Geodesic
Schrödinger 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 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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},
     year = {2003},
     volume = {295},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a8/}
}
TY  - JOUR
AU  - R. G. Shterenberg
TI  - Schrödinger operator in a periodic waveguide on the plane and quasi-conformal mappings
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2003
SP  - 204
EP  - 243
VL  - 295
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a8/
LA  - ru
ID  - ZNSL_2003_295_a8
ER  - 
%0 Journal Article
%A R. G. Shterenberg
%T Schrödinger operator in a periodic waveguide on the plane and quasi-conformal mappings
%J Zapiski Nauchnykh Seminarov POMI
%D 2003
%P 204-243
%V 295
%U http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a8/
%G ru
%F ZNSL_2003_295_a8
R. G. Shterenberg. Schrödinger 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/

[1] L. Ahlfors, L. Bers, “Riemann's mapping theorem for variable metrics”, Annals of Math., 72:2 (1960), 385–404 | DOI | MR | Zbl

[2] M. Sh. Birman, T. A. Suslina, “Dvumernyi periodicheskii magnitnyi gamiltonian absolyutno nepreryven”, Algebra i Analiz, 9:1 (1997), 32–48 | MR | Zbl

[3] M. Sh. Birman, T. A. Suslina, “Absolyutnaya nepreryvnost dvumernogo periodicheskogo magnitnogo gamiltoniana s razryvnym vektornym potentsialom”, Algebra i Analiz, 10:4 (1998), 1–36 | MR | Zbl

[4] M. Sh. Birman, T. A. Suslina, “Periodicheskii magnitnyi gamiltonian s peremennoi metrikoi. Problema absolyutnoi nepreryvnosti”, Algebra i Analiz, 11:2 (1999), 1–40 | MR | Zbl

[5] M. Sh. Birman, T. A. Suslina, R. G. Shterenberg, “Absolyutnaya nepreryvnost dvumernogo operatora Shredingera s delta-potentsialom, sosredotochennym na periodicheskoi sisteme krivykh”, Algebra i Analiz, 12:6 (2000), 140–177 | MR

[6] G. M. Goluzin, Geometricheskaya teoriya funktsii kompleksnogo peremennogo, GITTL, M., 1952 | MR

[7] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | Zbl

[8] V. G. Mazya, Prostranstva S. L. Soboleva, Izd-vo Leningr. un-ta, 1985 | MR

[9] A. Morame, “Absence of singular spectrum for a perturbation of a two-dimensional Laplace–Beltrami operator with periodic electromagnetic potential”, J. Phys. A, 31 (1998), 7593–7601 | DOI | MR | Zbl

[10] M. Z. Solomyak, “Spectral problems related to the critical exponent in the Sobolev embedding theorem”, Proc. London Math. Soc. (3), 71 (1995), 53–75 | DOI | MR | Zbl

[11] R. G. Shterenberg, “Absolyutnaya nepreryvnost dvumernogo magnitnogo periodicheskogo operatora Shredingera s elektricheskim potentsialom tipa proizvodnoi ot mery”, Zap. nauchn. semin. POMI, 271, 2000, 276–312 | MR | Zbl

[12] R. G. Shterenberg, “Absolyutnaya nepreryvnost spektra dvumernogo periodicheskogo operatora Shredingera s polozhitelnym elektricheskim potentsialom”, Algebra i Analiz, 13:4 (2001), 196–228 | MR | Zbl

[13] R. G. Shterenberg, “Absolyutnaya nepreryvnost spektra dvumernogo magnitnogo periodicheskogo operatora Shredingera s polozhitelnym elektricheskim potentsialom”, Trudy SPb mat. ob-va, 9, 2001, 199–233 | Zbl

[14] E. Shargorodsky, A. V. Sobolev, Quasi-conformal mappings and periodic spectral problems, University of Sussex at Brighton, Preprint No 2001-07, 2001

[15] A. V. Sobolev, J. Walthoe, Absolute continuity in periodic waveguides, University of Sussex at Brighton, Preprint No 2000-19, 2000 | MR

[16] T. A. Suslina, R. G. Shterenberg, “Absolyutnaya nepreryvnost spektra magnitnogo operatora Shredingera s metrikoi v dvumernom periodicheskom volnovode”, Algebra i Analiz, 14:2 (2002), 159–206 | MR | Zbl