具有奇异性的周期椭圆算子的绝对连续性

Abstract

Abstract:

In this paper we consider the spectral properties of periodic elliptic operator ∑^d_j,l₌₁D_jw(x)a_jlD_l+V(x) in R^d,d ≥ 3, where A=(a_jl) is a d×d positive definite matrix with real constant entries, V(x) and w(x) are periodic scalar function with respect to the same lattice, and w({x) is positive. Using a new uniform Sobolev inequalities on the d-torus established in [22], we prove that the spectrum of the operator is purely absolutely continuous if L_loc^2pd/d+2p(R^d) and w ∈ ∧_1+α^p, ∞(T^d)∩L^∞(T^d) for some α>0, p≥ d, or V ∈ L _loc^2d/3(R^d), w ∈ C¹(T^d), or V ∈ L_loc^d/2(R^d), w ∈ L_2, loc^d/2(T^d).

Key words: Elliptic operator, Periodic potential, Absolute continuous spectrum, Uniform Sobolev inequalities

CLC Number:

35J10

ZHAO Pei-Hao, LIU Wu-Long. On Absolute Continuity of Periodic Elliptic Operators with Singularity[J].Acta mathematica scientia,Series A, 2010, 30(1): 18-30.

Trendmd

References

[1] Birman M Sh, Suslina T A. Two-dimensional periodic magnetic Hamiltonian is absolutely continuous. Algebra i Analiz, 1997, 9(1): 32--48; Petersburg Math J, 1998, 9(1): 21--32

[2] Birman M Sh, Suslina T A. Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector-valued potential. Algebra i Analiz, 1998, 10(4): 1--36; Petersburg Math J, 1999, 10}(4)

[3] Birman M Sh, Suslina T A. A periodic magnetic Hamiltonian with a variable metric: The problem of absolute continuity. Algebra i Analiz, 1999, 11(2): 1--40; Petersburg Math J, 2000, 11(2)

[4] Birman M Sh, Suslina T A. On the absolute continuity of the periodic Schr\"{o}dinger and Dirac operators with magnetic potential. Differential Equations and Mathematical Physics, 1999: 41--49

[5] Danilov L. The spectrum of the two-dimensional periodic Schr\"{o}dinger operator. Teoret and Math Phys, 2003, 134(3): 392--403

[6] Danilov L. On the absolute continuity of the spectrum of a periodic Schr\"{o}dinger operator. Mat Zametki, 2003, 73(1): 49--62; Math Notes, 2003, 73(1/2): 46--57

[7] Friedlander L. On the spectrum of a class of second order perodic elliptic differential operators. Commun Math Phys, 2002, 229: 49--55

[8] Hempel R, Herbst I. Strong magnetic fields, Dirichlet boundaries, and spectral gaps. Comm Math Phys, 1995, 169: 237--259

[9] Kato T. Perturbation Theory for Linear Operators. New York: Springer-Verlag, 1976

[10] Kenig C E. Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique con-tinuation. Lecture Notes in Math, 1989, 1384: 69--90

[11] Kenig C E, Ruiz A, Sogge C D. Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math J, 1987, 55(2): 329--347

[12] Kuchment P. Floquet theory for partial differential equations. Berlin: Birkh\"{a}user Verlag, 1993

[13] Kuchment P, Levendorskii S. On the structure of spectra of periodic elliptic operators. Trans AMS, 2001, 354(2): 537--569

[14] Lapin I S. Absolute continuity of the spectra of two-dimensional periodic magnetic Schrdinger operator and Dirac operator with potentials in the Zygmund class, function theory and phase transitions. J Math Sci, 2001, 106(3): 2952--2974

[15] 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, 31: 7593--7601

[16] Reed M, Simon B. Methods of Modern Mathematical Physics. San Diego: Academic Press, 1978

[17] Saloff-coste L. Aspects of Sobolev-type Inequalities. Cambridge: Cambridge University Press, 2002

[18] Shen Z. On absolute continuity of the periodic Schr\"{o}dinger operators. Internat Math Res Notices, 2001, 2001(1): 1--31

[19] Shen Z. Absolute continuity of generalized periodic Schr\"{o}dinger operators. Contemp Math, 2001, 277: 113--126

[20] Shen Z. Absolute continuity of periodic Schr\"{o}dinger operators with potentials in the Kato class. Illinois J Math, 2001, 45(3): 873--893

[21] Shen Z. The periodic Schr\"{o}dinger operator with potentials in the Morrey class. Journal of Functional Analysis, 2002, 193: 314--345

[22] Shen Z, Zhao P H. Uniform Sobolev inequality and absolute continuity of periodic operators. Transactions of AMS, 2008, 360: 1741--1758

[23] Shterenberg R G. Absolute continuity of a two-dimensional magnetic periodic Schr\"{o}dinger operator with electric potential of measure derivative type. J Math Sci, 2003, 115(6): 2862--2882

[24] Shterenberg R G. Absolute continuity of the spectrum of the two-dimensional periodic Schr\"{o}dinger operator with strongly subordinate magnetic potential. Issled po Linein Oper i Teor Funkts, 2003, 31: 279--320

[25] Shterenberg R G. Absolute continuity of the spectrum of the two-dimensional magnetic periodic Schr\"{o}dinger operator with positive electric potential. Petersburg Mathematical Society IX, 2003: 191--221

[26] Sobolev A V. Absolute continuity of the periodic magnetic Schr\"{o}dinger operator. Invent Math, 1999, 137(1): 85--112

[27] Sogge C D. Concerning the L^p norm of spectral clusters of second order elliptic operators on compact manifolds. J Funct Analysis, 1988, 77(1): 123--138

[28] Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton Univ Press, 1970

[29] Suslina T. Absolute continuity of the spectrum of periodic operators of mathematical physics. Journées Equations aux Dérivées Partielle, 2000: 1--13

[30] Suslina T A, Shterenberg R G. Absolute continuity of the spectrum of the Schr\"{o}dinger operator with the potential concentrated on a periodic system of hyper-surfaces. Petersburg Math J, 2002, 13(5): 859--891

[31] Suslina T A, Shterenberg R G. Absolute continuity of the spectrum of the magnetic Schr\"{o}dinger operator with a metric in a two-dimensional periodic waveguide. Algebra i Analiz, 2003, 14(2): 305--343

[32] Thomas L E. Time dependent approach to scattering from impurities in a crystal. Comm Math Phys, 1973, 33: 335--343

[33] Thomas H, Hajo L, Peter M, Simone W. The absolute continuity of the integrated density of states for magnetic Schr\"{o}dinger operators with certain unbounded random potentials. Comm Math Phys, 2001, 221(2): 229--254

[34] Tikhomirov M, Filonov N. Absolute continuity of an "even'' periodic Schr\"{o}dinger operator with non-smooth coefficients. Algebra i Analiz, 2004, 16(3): 201--210

[35] Wolff T. A property of measures in Rⁿ and an application to unique continuation. Geom Funct Anal, 1992, 2: 225--284

On Absolute Continuity of Periodic Elliptic Operators with Singularity

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 4

Metrics

Comments

Recommended 10

[1]	DU Feng, WU Chuan-Xi, LI Guang-Han. Estimates for Eigenvalues of the Elliptic Operator in Divergence Form on the Heisenberg Group [J]. Acta mathematica scientia,Series A, 2012, 32(6): 1032-1040.
[2]	ZHENG Shen-Zhou, WANG Xi-Fen. Regularity of Very Weak Solutions for a Class of Quasilinear Subelliptic Equations [J]. Acta mathematica scientia,Series A, 2010, 30(2): 432-439.
[3]	LU Han-Fang, ZHENG Shen-Zhou. The Method of Green Function for a Holder Continuity of Elliptic Equations with Divergence [J]. Acta mathematica scientia,Series A, 2009, 29(5): 1160-1166.
[4]	JIN Yong-Yang, DAI Xin-Rong. The Continuity of the Solution for a Class of Degenerate Schrodinger Equations [J]. Acta mathematica scientia,Series A, 2004, 24(2): 238-245.