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Russia Izhevsk
Year
2014
Issue
2
Pages
3-28
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Section Mathematics
Title On the spectrum of a two-dimensional generalized periodic Schrödinger operator. II
Author(-s) Danilov L.I.a
Affiliations Physical Technical Institute, Ural Branch of the Russian Academy of Sciencesa
Abstract The paper is concerned with the problem of absolute continuity of the spectrum of the two-dimensional generalized periodic Schrödinger operator $H_g+V=-\nabla g\nabla +V$ where the continuous positive function $g$ and the scalar potential $V$ have a common period lattice $\Lambda $. The solutions of the equation $(H_g+V)\varphi =0$ determine, in particular, the electric field and the magnetic field of electromagnetic waves propagating in two-dimensional photonic crystals. The function $g$ and the scalar potential $V$ are expressed in terms of the electric permittivity $\varepsilon $ and the magnetic permeability $\mu $ ($V$ also depends on the frequency of the electromagnetic wave). The electric permittivity $\varepsilon $ may be a discontinuous function (and usually it is chosen to be piecewise constant) so the problem to relax the known smoothness conditions on the function $g$ that provide absolute continuity of the spectrum of the operator $H_g+V$ arises. In the present paper we assume that the Fourier coefficients of the functions $g^\pm \frac 12$ for some $q\in [1,\frac 43 )$ satisfy the condition $\sum \bigl( |N|^\frac 12|(g^\pm \frac 12)_N|\bigr) ^q < +\infty $, and the scalar potential $V$ has relative bound zero with respect to the operator $-\Delta $ in the sense of quadratic forms. Let $K$ be the fundamental domain of the lattice $\Lambda $, and assume that $K^*$ is the fundamental domain of the reciprocal lattice $\Lambda ^*$. The operator $H_g+V$ is unitarily equivalent to the direct integral of operators $H_g(k)+V$, with quasimomenta $k\in 2\pi K^*$, acting on the space $L^2(K)$. The last operators can be also considered for complex vectors $k+ik^\prime \in \mathbb C^2$. We use the Thomas method. The proof of absolute continuity of the spectrum of the operator $H_g+V$ amounts to showing that the operators $H_g(k+ik^\prime )+V-\lambda $, $\lambda \in \mathbb R$, are invertible for some appropriately chosen complex vectors $k+ik^\prime \in \mathbb C^2$ (depending on $g$, $V$, and the number $\lambda $) with sufficiently large imaginary parts $k^\prime $.
Keywords generalized Schrödinger operator, absolute continuity of the spectrum, periodic potential
UDC 517.958, 517.984.5
MSC 35P05
DOI 10.20537/vm140201
Received 28 February 2014
Language Russian
Citation Danilov L.I. On the spectrum of a two-dimensional generalized periodic Schrödinger operator. II, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 2, pp. 3-28.
References
  1. Reed M., Simon B. Methods of modern mathematical physics, Vol. II: Fourier analysis. Self-adjointness, New York: Academic, 1975. Translated under the title Metody sovremennoi matematicheskoi fiziki. II. Garmonicheskii analiz. Samosopryazhennost', Moscow: Mir, 1978, 400 p.
  2. Morame A. Absence of singular spectrum for a perturbation of a two-dimensional Laplace-Beltrami operator with periodic electro-magnetic potential, J. Phys. A: Math. Gen., 1998, vol. 31, pp. 7593-7601.
  3. Birman M.Sh., Suslina T.A. Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector valued potential, St. Petersburg Math. J., 1999, vol. 10, no. 4, pp. 579-601.
  4. Birman M.Sh., Suslina T.A. Periodic magnetic Hamiltonian with variable metric. The problem of absolute continuity, St. Petersburg Math. J., 2000, vol. 11, no. 2, pp. 203-232.
  5. Kuchment P., Levendorskii S. On the structure of spectra of periodic elliptic operators, Trans. Amer. Math. Soc., 2002, vol. 354, no. 2, pp. 537-569.
  6. Birman M.Sh., Suslina T.A., Shterenberg R.G. Absolute continuity of the two-dimensional Schrödinger operator with delta potential concentrated on a periodic system of curves, St. Petersburg Math. J., 2001, vol. 12, no. 6, pp. 983-1012.
  7. Shterenberg R.G. Absolute continuity of the spectrum of the two-dimensional magnetic periodic Schrödinger operator with positive electric potential, Trudy S.-Peterburg. Mat. Obshch., 2001, vol. 9, pp. 199-233 (in Russian).
  8. Shterenberg R.G. Absolute continuity of the spectrum of the two-dimensional periodic Schrödinger operator with positive electric potential, St. Petersburg Math. J., 2002, vol. 13, no. 4, pp. 659-683.
  9. Shterenberg R.G. Absolute continuity of the spectrum of the two-dimensional periodic Schrödinger operator with strongly subordinate magnetic potential, J. Math. Sci., 2005, vol. 129, pp. 4087-4109.
  10. Danilov L.I. On the spectrum of two-dimensional periodic Schrödinger and Dirac operators, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2002, no. 3 (26), pp. 3-98 (in Russian).
  11. Danilov L.I. On the spectrum of a two-dimensional periodic Schrödinger operator, Theor. Math. Phys., 2003, vol. 134, no. 3, pp. 392-403.
  12. Danilov L.I. On the absence of eigenvalues in the spectrum of two-dimensional periodic Dirac and Schrödinger operators, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2004, no. 1 (29), pp. 49-84 (in Russian).
  13. Danilov L.I. Absence of eigenvalues for the generalized two-dimensional periodic Dirac operator, St. Petersburg Math. J., 2006, vol. 17, no. 3, pp. 409-433.
  14. Danilov L.I. On the spectrum of a two-dimensional generalized periodic Schrödinger operator, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2013, no. 1 (41), pp. 79-96 (in Russian).
  15. Thomas L.E. Time dependent approach to scattering from impurities in a crystal, Commun. Math. Phys., 1973, vol. 33, pp. 335-343.
  16. Sobolev A.V. Absolute continuity of the periodic magnetic Schrödinger operator, Invent. Math., 1999, vol. 137, pp. 85-112.
  17. Shen Z. Absolute continuity of generalized periodic Schrödinger operators, Contemp. Math., 2001, vol. 277, pp. 113-126.
  18. Shen Z. The periodic Schrödinger operators with potentials in the Morrey class, J. Funct. Anal., 2002, vol. 193, pp. 314-345.
  19. Friedlander L. On the spectrum of a class of second order periodic elliptic differential operators, Commun. Math. Phys., 2002, vol. 229, pp. 49-55.
  20. Tikhomirov M., Filonov N. Absolute continuity of the “even’’ periodic Schrödinger operator with nonsmooth coefficients, St. Petersburg Math. J., 2005, vol. 16, no. 4, pp. 583-589.
  21. Shen Z., Zhao P. Uniform Sobolev inequalities and absolute continuity of periodic operators, Trans. Amer. Math. Soc., 2008, vol. 360, no. 4, pp. 1741-1758.
  22. Danilov L.I. On absolute continuity of the spectrum of a periodic magnetic Schrödinger operator, J. Phys. A: Math. Theor., 2009, vol. 42, 275204.
  23. Danilov L.I. On absolute continuity of the spectrum of three- and four-dimensional periodic Schrödinger operators, J. Phys. A: Math. Theor., 2010, vol. 43, 215201.
  24. Zhao P., Liu W. On absolute continuity of periodic elliptic operators with singularity, Acta Mathematica Scientia., 2010, vol. 30A, no. 1, pp. 18-30 (in Chinese).
  25. Danilov L.I. On the spectrum of a periodic Schrödinger operator with potential in the Morrey space, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, no. 3, pp. 25-47 (in Russian).
  26. Krupczyk K., Uhlmann G. Absolute continuity of the periodic Schrödinger operator in transversal geometry, 2013, http://arxiv.org/abs/1312.2989
  27. Kato T. Perturbation theory for linear operators, Berlin: Springer, 1966. Translated under the title Teoriya vozmushchenii lineinykh operatorov, Moscow: Mir, 1972.
  28. Danilov L.I. The spectrum of the Dirac operator with periodic potential. VI, Physical Technical Institute of the Ural Branch of the Russian Academy of Sciences, Izhevsk, 1996, 45 p. Deposited in VINITI 31.12.1996, no. 3855-B96 (in Russian).
  29. Filonov N., Sobolev A.V. Absence of the singular continuous component in the spectrum of analytic direct integrals, J. Math. Sci., 2006, vol. 136, pp. 3826-3831.
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