On moments of negative eigenvalues for the Pauli operator

Zhongwei Shen

doi:10.1006/jdeq.1998.3483

On moments of negative eigenvalues for the Pauli operator

Zhongwei Shen

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

4 Scopus citations

Abstract

This paper concerns the three-dimensional Pauli operator P=(σ·(p-A(x)))²+V(x) with a non-homogeneous magnetic field B=curl A. The following Lieb-Thirring type inequality for the moment of negative eigenvalues is established, ∑λ_j<0λ_j≤C₁∫_R³V(x)^5/2_-dx+C2∫_R³[b_p(x)]^3/2V(x)_-dx where p>3/2 and b_p(x) is the L^p average of B over a certain cube centered at x with a side length scaling like B^-1/2. We also show that, ifBhas a constant direction, ∑λ_j<0λ_j^γ≤C_1,γ∫_R³V(x)^γ+3/2_-dx+C_2,γ∫_R³b_p(x)V(x)^γ+1/2_-dx where γ>1/2 and p>1.

Original language	English
Pages (from-to)	420-455
Number of pages	36
Journal	Journal of Differential Equations
Volume	151
Issue number	2
DOIs	https://doi.org/10.1006/jdeq.1998.3483
State	Published - Jan 20 1999

Bibliographical note

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

Funding

* Research supported in part by the AMS Centennial Research Berkeley, California, and the NSF grant DMS-9596266.

Funders	Funder number
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	DMS-9596266

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1006/jdeq.1998.3483

Cite this

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title = "On moments of negative eigenvalues for the Pauli operator",

abstract = "This paper concerns the three-dimensional Pauli operator P=(σ·(p-A(x)))2+V(x) with a non-homogeneous magnetic field B=curl A. The following Lieb-Thirring type inequality for the moment of negative eigenvalues is established, ∑λj<0λj≤C1∫R3V(x)5/2-dx+C2∫R3[bp(x)]3/2V(x)-dx where p>3/2 and bp(x) is the Lp average of B over a certain cube centered at x with a side length scaling like B-1/2. We also show that, ifBhas a constant direction, ∑λj<0λjγ≤C1,γ∫R3V(x)γ+3/2-dx+C2,γ∫R3bp(x)V(x)γ+1/2-dx where γ>1/2 and p>1.",

author = "Zhongwei Shen",

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

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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 Berkeley, California, and the NSF grant DMS-9596266.

PY - 1999/1/20

Y1 - 1999/1/20

N2 - This paper concerns the three-dimensional Pauli operator P=(σ·(p-A(x)))2+V(x) with a non-homogeneous magnetic field B=curl A. The following Lieb-Thirring type inequality for the moment of negative eigenvalues is established, ∑λj<0λj≤C1∫R3V(x)5/2-dx+C2∫R3[bp(x)]3/2V(x)-dx where p>3/2 and bp(x) is the Lp average of B over a certain cube centered at x with a side length scaling like B-1/2. We also show that, ifBhas a constant direction, ∑λj<0λjγ≤C1,γ∫R3V(x)γ+3/2-dx+C2,γ∫R3bp(x)V(x)γ+1/2-dx where γ>1/2 and p>1.

AB - This paper concerns the three-dimensional Pauli operator P=(σ·(p-A(x)))2+V(x) with a non-homogeneous magnetic field B=curl A. The following Lieb-Thirring type inequality for the moment of negative eigenvalues is established, ∑λj<0λj≤C1∫R3V(x)5/2-dx+C2∫R3[bp(x)]3/2V(x)-dx where p>3/2 and bp(x) is the Lp average of B over a certain cube centered at x with a side length scaling like B-1/2. We also show that, ifBhas a constant direction, ∑λj<0λjγ≤C1,γ∫R3V(x)γ+3/2-dx+C2,γ∫R3bp(x)V(x)γ+1/2-dx where γ>1/2 and p>1.

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DO - 10.1006/jdeq.1998.3483

M3 - Article

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SN - 0022-0396

VL - 151

SP - 420

EP - 455

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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