Uniform sobolev inequalities and absolute continuity of periodic operators

Zhongwei Shen; Peihao Zhao

doi:10.1090/S0002-9947-07-04545-X

Uniform sobolev inequalities and absolute continuity of periodic operators

Zhongwei Shen, Peihao Zhao

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

Abstract

We establish certain uniform L^p- L^q inequalities for a family of second order elliptic operators of the form (D + k)A(D + k) ^T on the d-torus, where D =-i∇, k ∈ ℂ^d and A is a symmetric, positive definite d × 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 ℝ^d with singular potential. The absolute continuity of the elliptic operator div(ω(x)∇) on ℝ^d with a positive periodic scalar function ω(x) is also studied.

Original language	English
Pages (from-to)	1741-1758
Number of pages	18
Journal	Transactions of the American Mathematical Society
Volume	360
Issue number	4
DOIs	https://doi.org/10.1090/S0002-9947-07-04545-X
State	Published - Apr 2008

Bibliographical note

Funding Information:
We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Science and Culture of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Sci- ence Council of the Republic of China; the Foundation; and the Max Kade Foundation.

Keywords

Absolute continuous spectrum
Dirac operator
Periodic potential
Uniform Sobolev inequalities

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1090/S0002-9947-07-04545-X

Cite this

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title = "Uniform sobolev inequalities and absolute continuity of periodic operators",

abstract = "We establish certain uniform Lp- Lq inequalities for a family of second order elliptic operators of the form (D + k)A(D + k) T on the d-torus, where D =-i∇, k ∈ ℂd and A is a symmetric, positive definite d × 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 ℝd with singular potential. The absolute continuity of the elliptic operator div(ω(x)∇) on ℝd with a positive periodic scalar function ω(x) is also studied.",

keywords = "Absolute continuous spectrum, Dirac operator, Periodic potential, Uniform Sobolev inequalities",

author = "Zhongwei Shen and Peihao Zhao",

note = "Funding Information: We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Science and Culture of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Sci- ence Council of the Republic of China; the Foundation; and the Max Kade Foundation.",

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AU - Shen, Zhongwei

AU - Zhao, Peihao

N1 - Funding Information: We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Science and Culture of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Sci- ence Council of the Republic of China; the Foundation; and the Max Kade Foundation.

PY - 2008/4

Y1 - 2008/4

N2 - We establish certain uniform Lp- Lq inequalities for a family of second order elliptic operators of the form (D + k)A(D + k) T on the d-torus, where D =-i∇, k ∈ ℂd and A is a symmetric, positive definite d × 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 ℝd with singular potential. The absolute continuity of the elliptic operator div(ω(x)∇) on ℝd with a positive periodic scalar function ω(x) is also studied.

AB - We establish certain uniform Lp- Lq inequalities for a family of second order elliptic operators of the form (D + k)A(D + k) T on the d-torus, where D =-i∇, k ∈ ℂd and A is a symmetric, positive definite d × 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 ℝd with singular potential. The absolute continuity of the elliptic operator div(ω(x)∇) on ℝd with a positive periodic scalar function ω(x) is also studied.

KW - Absolute continuous spectrum

KW - Dirac operator

KW - Periodic potential

KW - Uniform Sobolev inequalities

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DO - 10.1090/S0002-9947-07-04545-X

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VL - 360

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JO - Transactions of the American Mathematical Society

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ER -

Uniform sobolev inequalities and absolute continuity of periodic operators

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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