Absolute continuity of the “even" periodic Schrödinger operator with nonsmooth coefficients
HTML articles powered by AMS MathViewer
- by
M. Tikhomirov and N. Filonov
Translated by: M. Tikhomirov - St. Petersburg Math. J. 16 (2005), 583-589
- DOI: https://doi.org/10.1090/S1061-0022-05-00866-6
- Published electronically: May 2, 2005
- PDF | Request permission
References
- M. Sh. Birman and M. Z. Solomyak, Schrödinger operator. Estimates for number of bound states as function-theoretical problem, Spectral theory of operators (Novgorod, 1989) Amer. Math. Soc. Transl. Ser. 2, vol. 150, Amer. Math. Soc., Providence, RI, 1992, pp. 1–54. MR 1157648, DOI 10.1090/trans2/150/01
- M. Sh. Birman and T. A. Suslina, Absolute continuity of a two-dimensional periodic magnetic Hamiltonian with discontinuous vector potential, Algebra i Analiz 10 (1998), no. 4, 1–36 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 10 (1999), no. 4, 579–601. MR 1654063
- M. Sh. Birman and T. A. Suslina, A periodic magnetic Hamiltonian with a variable metric. The problem of absolute continuity, Algebra i Analiz 11 (1999), no. 2, 1–40 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 11 (2000), no. 2, 203–232. MR 1702587
- A. N. Kolmogorov and S. V. Fomin, Èlementy teorii funktsiĭ i funktsional′nogo analiza, 6th ed., “Nauka”, Moscow, 1989 (Russian). With a supplement, “Banach algebras”, by V. M. Tikhomirov. MR 1025126
- Herbert Koch and Daniel Tataru, Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients, Comm. Pure Appl. Math. 54 (2001), no. 3, 339–360. MR 1809741, DOI 10.1002/1097-0312(200103)54:3<339::AID-CPA3>3.0.CO;2-D
- Abderemane Morame, Absence of singular spectrum for a perturbation of a two-dimensional Laplace-Beltrami operator with periodic electromagnetic potential, J. Phys. A 31 (1998), no. 37, 7593–7601. MR 1652918, DOI 10.1088/0305-4470/31/37/017
- Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
- Alexander V. Sobolev, Absolute continuity of the periodic magnetic Schrödinger operator, Invent. Math. 137 (1999), no. 1, 85–112. MR 1703339, DOI 10.1007/s002220050324
- T. A. Suslina and R. G. Shterenberg, Absolute continuity of the spectrum of the Schrödinger operator with the potential concentrated on a periodic system of hypersurfaces, Algebra i Analiz 13 (2001), no. 5, 197–240 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 5, 859–891. MR 1882869
- Lawrence E. Thomas, Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys. 33 (1973), 335–343. MR 334766
- N. Filonov, Second-order elliptic equation of divergence form having a compactly supported solution, J. Math. Sci. (New York) 106 (2001), no. 3, 3078–3086. Function theory and phase transitions. MR 1906035, DOI 10.1023/A:1011379807662
- L. Friedlander, On the spectrum of a class of second order periodic elliptic differential operators, Comm. Math. Phys. 229 (2002), no. 1, 49–55. MR 1917673, DOI 10.1007/s00220-002-0675-6
- R. Shterenberg, Absolute continuity of the spectrum of a two-dimensional periodic magnetic Schrödinger operator with positive electric potential, Trudy S.-Peterburg. Mat. Obshch. 9 (2001), 199–233. (Russian)
- Friedmar Schulz, On the unique continuation property of elliptic divergence form equations in the plane, Math. Z. 228 (1998), no. 2, 201–206. MR 1630571, DOI 10.1007/PL00004610
Bibliographic Information
- M. Tikhomirov
- Affiliation: Department of Physics, St. Petersburg State University, Ulyanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia
- Email: misha@mt5788.spb.edu
- N. Filonov
- Affiliation: Department of Physics, St. Petersburg State University, Ulyanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia
- MR Author ID: 609754
- Email: filonov@mph.phys.spbu.ru
- Received by editor(s): September 1, 2003
- Published electronically: May 2, 2005
- Additional Notes: The first author was supported by RFBR (grants nos. 01–01–00218 and 02–01–00798)
- © Copyright 2005 American Mathematical Society
- Journal: St. Petersburg Math. J. 16 (2005), 583-589
- MSC (2000): Primary 35Q40
- DOI: https://doi.org/10.1090/S1061-0022-05-00866-6
- MathSciNet review: 2083570