Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Uniform Sobolev inequalities and absolute continuity of periodic operators


Authors: Zhongwei Shen and Peihao Zhao
Journal: Trans. Amer. Math. Soc. 360 (2008), 1741-1758
MSC (2000): Primary 35J10, 42B15
DOI: https://doi.org/10.1090/S0002-9947-07-04545-X
Published electronically: November 26, 2007
MathSciNet review: 2366961
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We establish certain uniform $ L^{p}-L^{q}$ inequalities for a family of second order elliptic operators of the form $ ( {\bold {D}} + {\bold {k}} ) A ( {\bold {D}}+ {\bold {k} })^{T}$ on the $ d$-torus, where $ {\bold {D}} =-i\nabla , {\bold {k}}\in {\Bbb {C}} ^{d}$ and $ A$ is a symmetric, positive definite $ d\times d$ matrix with real constant entries. Using these Sobolev type inequalities, we obtain the absolute continuity of the spectrum of the periodic Dirac operator on $ {\Bbb R}^{d}$ with singular potential. The absolute continuity of the elliptic operator div $ (\omega ( {\bold {x}})\nabla )$ on $ {\Bbb R}^{d}$ with a positive periodic scalar function $ \omega ( {\bold {x}} )$ is also studied.


References [Enhancements On Off] (What's this?)

  • [1] M.Sh. Birman and T.A. Suslina, Two-dimensional periodic magnetic Hamiltonian is absolutely continuous, Algebra i Analiz 9 (1) (1997), 32-48; English transl. in St. Petersburg Math. J. 9 (1) (1998), 21-32. MR 1458417 (98g:47038)
  • [2] 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 (4), 1-36; English transl. in St. Petersburg Math. J. 10 (4) (1999), 579-601. MR 1654063 (99k:81060)
  • [3] M.Sh. Birman and T.A. Suslina, The periodic Dirac operator is absolutely continuous, Integral Equations Operator Theory 34 (4) (1999), 377-395. MR 1702229 (2000h:47068)
  • [4] M.Sh. Birman and T.A. Suslina, A periodic magnetic Hamiltonian with a variable metric: The problem of absolute continuity, Algebra i Analiz 11 (2) (1999), 1-40; English transl. in. St. Petersburg Math. J. 11 (2) (2000), 203-232. MR 1702587 (2000i:35026)
  • [5] M.Sh. Birman, T.A. Suslina, and R.G. Shterenberg, Absolute continuity of the two-dimensional Schrödinger operator with potential supported on a periodic system of curves, Algebra i Analiz 12 (6) (2000), 140-177; English transl. in. St. Petersburg Math. J. 12 (6) (2001), 983-1012. MR 1816514 (2002k:35227)
  • [6] L.I. Danilov, On the spectrum of the Dirac operator with periodic potential in $ {\Bbb R}^{n}$, Teoret. Mat. Fiz. 85 (1) (1990), 41-53; English transl. in Theoret. Math. Phys. 85 (1) (1991), 1039-1048. MR 1083951 (92a:35119)
  • [7] L.I. Danilov, Resolvent estimates and the spectrum of the Dirac operator with a periodic potential, Teoret. Mat. Fiz 103 (1) (1995), 3-22; English transl. in. Theoret. Math. Phys. 103 (1) (1995), 349-365. MR 1470934 (98f:35112)
  • [8] L.I. Danilov, On the spectrum of the periodic Dirac operator, Teoret. Mat. Fiz. 124 (1) (2000), 3-17; English transl. in. Theoret. Math. Phys. 124 (1) (2000), 859-871. MR 1821309 (2002b:81028)
  • [9] L.I. Danilov, Absolute continuity of the spectrum of a periodic Dirac operator, Differ. Uravn. 36 (2) (2000), 233-240; English transl. in. Differ. Equ. 36 (2) (2000), 262-271. MR 1773794 (2001f:47082)
  • [10] L.I. Danilov, On the spectrum of the two-dimentional periodic Schrödinger operator, Teoret. Mat. Fiz. 134 (2003), 447-459; English transl. in Theoret. Math. Phys. 134 (2003), 392-403. MR 2001818 (2004j:35210)
  • [11] L.I. Danilov, On the absolute continuity of the spectrum of a periodic Schrödinger operator, Mat. Zametki 73 (2003), no. 1, 49-62; English transl. in Math. Notes 73 (1) (2003), 46-57. MR 1993539 (2004f:35130)
  • [12] L.I. Danilov, On absence of eigenvalues in the spectrum of generalized two-dimensional periodic Dirac operators, Algebra i Analiz 17 (3) (2005), 47-80. MR 2167843 (2006m:35261)
  • [13] L. Friedlander, On the spectrum of a class of second order periodic elliptic differential operators, Commun. Math. Phys. 229 (2002), 49-55. MR 1917673 (2003k:35179)
  • [14] L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., 2004.
  • [15] C.E. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation, Lecture Notes in Math. 1384 (1989), 69-90. MR 1013816 (90m:35016)
  • [16] C.E. Kenig, A. Ruiz, and C.D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 (2) (1987), 329-347. MR 894584 (88d:35037)
  • [17] P. Kuchment, Floquet Theory for Partial Differential Equations, Birkhäuser Verlag, 1993. MR 1232660 (94h:35002)
  • [18] P. Kuchment and S. Levendorskii, On the structure of spectra of periodic elliptic operators, Trans. Amer. Math. Soc. 354 (2) (2002), 537-569. MR 1862558 (2002g:35163)
  • [19] I.S. Lapin, Absolute continuity of the spectra of two-dimensional periodic magnetic Schrödinger operator and Dirac operator with potentials in the Zygmund class, Function theory and phase transitions. J. Math. Sci. (New York) 106 (3) (2001), 2952-2974. MR 1906028 (2003h:35182)
  • [20] 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 1652918 (99i:81039)
  • [21] M. Reed and B. Simon, Methods of Modern Mathematical Physics,, vol. IV, Academic Press, 1978. MR 0493422 (58:12430a)
  • [22] Z. Shen, On absolute continuity of the periodic Schrödinger operators, Internat. Math. Res. Notices 2001 (1) (2001), 1-31. MR 1809495 (2002a:47078)
  • [23] Z. Shen, Absolute continuity of generalized periodic Schrödinger operators, Contemp. Math. 277 (2001), 113-126. MR 1840430 (2002j:35078)
  • [24] Z. Shen, Absolute continuity of periodic Schrödinger operators with potentials in the Kato class, Illinois J. Math. 45 (3) (2001), 873-893. MR 1879241 (2002m:35036)
  • [25] Z. Shen, The periodic Schrödinger operator with potentials in the Morrey class, J. Funct. Anal. 193 (2002), 314-345. MR 1929505 (2003k:47071)
  • [26] R. G. Shterenberg, Absolute continuity of a two-dimensional magnetic periodic Schrödinger operator with electric potential of measure derivative type, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 31, 276-312, 318; English transl. in J. Math. Sci. (N.Y.) 115 (6) (2003), 2862-2882. MR 1810620 (2002m:35171)
  • [27] R. G. Shterenberg, Absolute continuity of the spectrum of the two-dimensional magnetic periodic Schrödinger operator with positive electric potential., Proceedings of the St. Petersburg Mathematical Society IX (2003), 191-221. MR 2018378 (2005d:35185)
  • [28] R. G. Shterenberg, Absolute continuity of the spectrum of the two-dimensional periodic Schrödinger operator with strongly subordinate magnetic potential (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 303 (2003). MR 2018378 (2005d:35185)
  • [29] R.G. Shterenberg and T.A. Suslina, Absolute continuity of the spectrum for the Schrödinger operator with the potential concentrated on a periodic system of hypersurfaces, St. Petersburg Math. J. 13 (2002), 859-891. MR 1882869 (2002m:35172)
  • [30] A.V. Sobolev, Absolute continuity of the periodic magnetic Schrödinger operator, Invent. Math. 137 (1) (1999), 85-112. MR 1703339 (2000g:35028)
  • [31] C.D. Sogge, Concerning the $ L^{p}$ norm of spectral clusters of second order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1) (1988), 123-138. MR 930395 (89d:35131)
  • [32] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. MR 0290095 (44:7280)
  • [33] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971. MR 0304972 (46:4102)
  • [34] T. Suslina, Absolute continuity of the spectrum of periodic operators of mathematical physics, Journées Equations aux Dérivées Partielles (2000), 1-13. MR 1775694 (2001f:35295)
  • [35] L.E. Thomas, Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys. 33 (1973), 335-343. MR 0334766 (48:13084)
  • [36] M. Tikhomirov and N. Filonov, Absolute continuity of an "even" periodic Schrödinger operator with non-smooth coefficients, Algebra i Analiz 16 (3) (2004), 201-210; English transl. in St. Petersburg Math. J. 16 (3) (2005), 583-589. MR 2083570 (2005f:35056)
  • [37] T. Wolff, A property of measures in $ \mathbb{R}^{n}$ and an application to unique continuation, Geom. Funct. Anal. 2 (1992), 225-284. MR 1159832 (93c:35015)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J10, 42B15

Retrieve articles in all journals with MSC (2000): 35J10, 42B15


Additional Information

Zhongwei Shen
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: shenz@ms.uky.edu

Peihao Zhao
Affiliation: Department of Mathematics, Lanzhou University, Lanzhou, Gansu, 730000, People’s Republic of China
Email: zhaoph@lzu.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-07-04545-X
Keywords: Dirac operator, periodic potential, absolute continuous spectrum, uniform Sobolev inequalities
Received by editor(s): July 13, 2005
Published electronically: November 26, 2007
Additional Notes: The first author was supported in part by the NSF (DMS-0500257). The second author was supported in part by the NSF of Gansu Province, China (ZS021-A25-002-Z) and the NSFC (10371052).
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society