On the spectrum of a two-dimensional generalized periodic Schrödinger operator. II

L. I. Danilov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 2 (2014), pp. 3-28

Résumé

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 $k^{\prime }$.

Détail

Keywords: generalized Schrödinger operator, absolute continuity of the spectrum, periodic potential.

Bibliographie

[1] Reed M., Simon B., Methods of modern mathematical physics, v. II, Fourier analysis. Self-adjointness, Academic, New York, 1975 | MR | MR | Zbl

[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., 31 (1998), 7593–7601 | DOI | MR | Zbl

[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., 10:4 (1999), 579–601 | MR | Zbl

[4] Birman M. Sh., Suslina T. A., “Periodic magnetic Hamiltonian with variable metric. The problem of absolute continuity”, St. Petersburg Math. J., 11:2 (2000), 203–232 | MR | Zbl

[5] Kuchment P., Levendorskiĭ S., “On the structure of spectra of periodic elliptic operators”, Trans. Amer. Math. Soc., 354:2 (2002), 537–569 | DOI | MR | Zbl

[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., 12:6 (2001), 983–1012 | MR | Zbl

[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., 9, 2001, 199–233 (in Russian) | Zbl

[8] Shterenberg R. G., “Absolute continuity of the spectrum of the two-dimensional periodic Schrödinger operator with positive electric potential”, St. Petersburg Math. J., 13:4 (2002), 659–683 | MR | Zbl

[9] Shterenberg R. G., “Absolute continuity of the spectrum of the two-dimensional periodic Schrödinger operator with strongly subordinate magnetic potential”, J. Math. Sci., 129 (2005), 4087–4109 | DOI | MR | Zbl

[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), 3–98 (in Russian) | MR

[11] Danilov L. I., “On the spectrum of a two-dimensional periodic Schrödinger operator”, Theor. Math. Phys., 134:3 (2003), 392–403 | DOI | DOI | MR | Zbl

[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), 49–84 (in Russian)

[13] Danilov L. I., “Absence of eigenvalues for the generalized two-dimensional periodic Dirac operator”, St. Petersburg Math. J., 17:3 (2006), 409–433 | DOI | MR | Zbl

[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), 79–96 (in Russian) | MR

[15] Thomas L. E., “Time dependent approach to scattering from impurities in a crystal”, Commun. Math. Phys., 33 (1973), 335–343 | DOI | MR

[16] Sobolev A. V., “Absolute continuity of the periodic magnetic Schrödinger operator”, Invent. Math., 137 (1999), 85–112 | DOI | MR | Zbl

[17] Shen Z., “Absolute continuity of generalized periodic Schrödinger operators”, Contemp. Math., 277, 2001, 113–126 | DOI | MR | Zbl

[18] Shen Z., “The periodic Schrödinger operators with potentials in the Morrey class”, J. Funct. Anal., 193 (2002), 314–345 | DOI | MR | Zbl

[19] Friedlander L., “On the spectrum of a class of second order periodic elliptic differential operators”, Commun. Math. Phys., 229 (2002), 49–55 | DOI | MR | Zbl

[20] Tikhomirov M., Filonov N., “Absolute continuity of the “even” periodic Schrödinger operator with nonsmooth coefficients”, St. Petersburg Math. J., 16:4 (2005), 583–589 | MR | Zbl

[21] Shen Z., Zhao P., “Uniform Sobolev inequalities and absolute continuity of periodic operators”, Trans. Amer. Math. Soc., 360:4 (2008), 1741–1758 | DOI | MR | Zbl

[22] Danilov L. I., “On absolute continuity of the spectrum of a periodic magnetic Schrödinger operator”, J. Phys. A: Math. Theor., 42 (2009), 275204 | DOI | MR | Zbl

[23] Danilov L. I., “On absolute continuity of the spectrum of three- and four-dimensional periodic Schrödinger operators”, J. Phys. A: Math. Theor., 43 (2010), 215201 | DOI | MR | Zbl

[24] Zhao P., Liu W., “On absolute continuity of periodic elliptic operators with singularity”, Acta Mathematica Scientia, 30A:1 (2010), 18–30 (in Chinese) | MR | Zbl

[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, 25–47 (in Russian) | MR

[26] Krupczyk K., Uhlmann G., Absolute continuity of the periodic Schrödinger operator in transversal geometry, arXiv: 1312.2989

[27] Kato T., Perturbation theory for linear operators, Springer, Berlin, 1966 | MR | Zbl | Zbl

[28] Danilov L. I., The spectrum of the Dirac operator with periodic potential, VI, Deposited in VINITI 31.12.1996, No 3855-B96, Physical Technical Institute of the Ural Branch of the Russian Academy of Sciences, Izhevsk, 1996, 45 pp. (in Russian)

[29] J. Math. Sci., 136 (2006), 3826–3831 | DOI | MR | Zbl