Absolute continuity of the spectrum of periodic operators of mathematical physics
Journées équations aux dérivées partielles (2000), article no. 18, 13 p.

The lecture is devoted to the problem of absolute continuity of the spectrum of periodic operators. A general approach to this problem was suggested by L. Thomas in 1973 for the case of the Schrödinger operator with periodic electric potential. Further application of his method to concrete operators of mathematical physics met analytic difficulties. In recent years several new problems in this area have been solved. We propose a survey of known results in this area, including very recent, and formulate unsolved problems.

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title = {Absolute continuity of the spectrum of periodic operators of mathematical physics},
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Suslina, Tatiana. Absolute continuity of the spectrum of periodic operators of mathematical physics. Journées équations aux dérivées partielles (2000), article  no. 18, 13 p. http://www.numdam.org/item/JEDP_2000____A18_0/

[BSu1] M. Sh. Birman and T. A. Suslina, Two-dimensional periodic magnetic Hamiltonian is absolutely continuous, Algebra i Analiz 9 (1997), no. 1, 32-48. English transl. St. Petersburg Math. J. 9 (1998), no. 1. | Zbl 0890.35096

[BSu2] M. Sh. Birman and T. A. Suslina, Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector-valued potential, Algebra i Analiz 10 (1998), no. 4, 1-36. English transl. St. Petersburg Math. J. 10 (1999), no. 4. | Zbl 0922.35101

[BSu3] M. Sh. Birman and T. A. Suslina, The periodic Dirac operator is absolutely continuous, Integral Equations and Operator Theory 34 (1999), no. 4, 377-395. | MR 2000h:47068 | Zbl 0937.35032

[BSu4] M. Sh. Birman and T. A. Suslina, Periodic magnetic Hamiltonian with variable metric. The problem of absolute continuity, Algebra i Analiz 11 (1999), no. 2, 1-40. English transl. St. Petersburg Math. J. 11 (2000), no. 2. | Zbl 0941.35015

[BSu5] M. Sh. Birman and T. A. Suslina, On the absolute continuity of the periodic Schrödinger and Dirac operators with magnetic potential, Diff. Eq. Math. Physics, Proc. of an Intern. Conf. held at the University of Alabama at Birmingham, March 16-20, 1999, 17-25.

[BSuSht] M. Sh. Birman, T. A. Suslina and R. G. Shterenberg, Absolute continuity of the spectrum of the Schrödinger operator with delta-like potential supported by a periodic graph, in preparation.

[D1] L. Danilov, On the spectrum of the Dirac operator with periodic potential in ℝn, Teor. Mat. Fiz. 85 (1990), no. 1, 41-53. | MR 92a:35119 | Zbl 0723.47043

[D2] L. Danilov, Resolvent estimates and the spectrum of the Dirac operator with periodic potential, Teor. Mat. Fiz. 103 (1995), no. 1, 3-22. | MR 98f:35112 | Zbl 0855.35105

[D3] L. Danilov, On the spectrum of the twodimensional periodic Dirac operator, Teor. Mat. Fiz. 118 (1999), no. 1, 3-14. | MR 2000h:35117 | Zbl 01466058

[D4] L. Danilov, Private communication (1999).

[HemHer] R. Hempel and I. Herbst, Bands and gaps for periodic magnetic Hamiltonians, Oper. Theory Adv. Appl., 78 (1995), 175-184. | MR 97d:35158 | Zbl 0849.35091

[K1] P. Kuchment, Floquet theory for partial differential equations, Birkhäuser, Basel, 1993. | MR 94h:35002 | Zbl 0789.35002

[K2] P. Kuchment, Private communication (2000).

[K3] P. Kuchment, The mathematics of photonic crystals, To appear in «Math. Modeling in Optical Science», SIAM. | Zbl 0986.78004

[KL] P. Kuchment and S. Levendorskii, On the absolute continuity of spectra of periodic elliptic operators, Oper. Theory Adv. Appl. 108 (1999), 291-297. | MR 2001c:35166 | Zbl 0965.35100

[M1] A. Morame, Absence of singular spectrum for a perturbation of a two-dimensional Laplace-Beltrami operator with periodic electro-magnetic potential, J. Phys. A: Math. Gen. 31 (1998), 7593-7601. | MR 99i:81039 | Zbl 0931.35143

[M2] A. Morame, The absolute continuity of the spectrum of Maxwell operator in a periodic media, Preprint 99-308 in mp-arc (1999).

[RSi] M. Reed and B. Simon, Methods of modern mathematical physics, IV, Academic Press, New York, 1975. | Zbl 0308.47002

[Sh1] Z. Shen, On absolute continuity of the periodic Schrödinger operators, Preprint ESI 597, Vienna (1998).

[Sh2] Z. Shen, On absolute continuity of the periodic Schrödinger operators, Preprint 99-189 in mp-arc (1999).

[Sh3] Z. Shen, The periodic Schrödinger operator with potentials in the C. Fefferman-Phong class, Preprint 99-455 in mp-arc (1999).

[So1] A. Sobolev, Absolute continuity of the periodic magnetic Schrödinger operator, Invent. Math. 137 (1997), 85-112. | MR 2000g:35028 | Zbl 0932.35049

[So2] A. Sobolev, Private communication (1998).

[T] L. Thomas, Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys. 33 (1973), 335-343. | MR 48 #13084