On Moments of Negative Eigenvalues for the Pauli Operator

Zhongwei Shen

doi:10.1006/jdeq.1998.3473

On Moments of Negative Eigenvalues for the Pauli Operator

Zhongwei Shen

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

10 Scopus citations

Abstract

This paper concerns the three-dimensional Pauli operatorP=(σ·(p-A(x)))²+V(x)with a nonhomogeneous magnetic fieldB=curlA. The following Lieb-Thirring type inequality for the moment of negative eigenvalues is established as∑λ_j<0λ_j≤C₁∫ _R³V(x)^5/2_-dx+C₂∫ _R³[b_p(x)]^3/2V(x) _-dx,wherep>3/2 andb_p(x) is theL^paverage of B over certain cube centered atxwith a side length scaling like B^-1/2. We also show that, ifBhas a constant direction,∑λ_j<0λ_j≤C _1,λ∫_R³V(x) ^γ+3/2_-dx+C_2,γ∫_R ³b_p(x)V(x)^γ+1/2 _-dx,whereγ>1/2 andp>1.

Original language	English
Pages (from-to)	292-327
Number of pages	36
Journal	Journal of Differential Equations
Volume	149
Issue number	2
DOIs	https://doi.org/10.1006/jdeq.1998.3473
State	Published - Nov 1 1998

Bibliographical note

Funding Information:
* Research supported in part by the AMS Centennial Research Berkely, California, and the NSF grant DMS-9596266.

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1006/jdeq.1998.3473

Cite this

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title = "On Moments of Negative Eigenvalues for the Pauli Operator",

abstract = "This paper concerns the three-dimensional Pauli operatorP=(σ·(p-A(x)))2+V(x)with a nonhomogeneous magnetic fieldB=curlA. The following Lieb-Thirring type inequality for the moment of negative eigenvalues is established as∑λj<0λj≤C1∫ R3V(x)5/2-dx+C2∫ R3[bp(x)]3/2V(x) -dx,wherep>3/2 andbp(x) is theLpaverage of B over certain cube centered atxwith a side length scaling like B-1/2. We also show that, ifBhas a constant direction,∑λj<0λj≤C 1,λ∫R3V(x) γ+3/2-dx+C2,γ∫R 3bp(x)V(x)γ+1/2 -dx,whereγ>1/2 andp>1.",

author = "Zhongwei Shen",

note = "Funding Information: * Research supported in part by the AMS Centennial Research Berkely, California, and the NSF grant DMS-9596266.",

year = "1998",

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doi = "10.1006/jdeq.1998.3473",

language = "English",

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T1 - On Moments of Negative Eigenvalues for the Pauli Operator

AU - Shen, Zhongwei

N1 - Funding Information: * Research supported in part by the AMS Centennial Research Berkely, California, and the NSF grant DMS-9596266.

PY - 1998/11/1

Y1 - 1998/11/1

N2 - This paper concerns the three-dimensional Pauli operatorP=(σ·(p-A(x)))2+V(x)with a nonhomogeneous magnetic fieldB=curlA. The following Lieb-Thirring type inequality for the moment of negative eigenvalues is established as∑λj<0λj≤C1∫ R3V(x)5/2-dx+C2∫ R3[bp(x)]3/2V(x) -dx,wherep>3/2 andbp(x) is theLpaverage of B over certain cube centered atxwith a side length scaling like B-1/2. We also show that, ifBhas a constant direction,∑λj<0λj≤C 1,λ∫R3V(x) γ+3/2-dx+C2,γ∫R 3bp(x)V(x)γ+1/2 -dx,whereγ>1/2 andp>1.

AB - This paper concerns the three-dimensional Pauli operatorP=(σ·(p-A(x)))2+V(x)with a nonhomogeneous magnetic fieldB=curlA. The following Lieb-Thirring type inequality for the moment of negative eigenvalues is established as∑λj<0λj≤C1∫ R3V(x)5/2-dx+C2∫ R3[bp(x)]3/2V(x) -dx,wherep>3/2 andbp(x) is theLpaverage of B over certain cube centered atxwith a side length scaling like B-1/2. We also show that, ifBhas a constant direction,∑λj<0λj≤C 1,λ∫R3V(x) γ+3/2-dx+C2,γ∫R 3bp(x)V(x)γ+1/2 -dx,whereγ>1/2 andp>1.

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DO - 10.1006/jdeq.1998.3473

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EP - 327

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -

On Moments of Negative Eigenvalues for the Pauli Operator

Abstract

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ASJC Scopus subject areas

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