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.

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.1006/jdeq.1998.3483

Cite this

@article{6e2e79f7060a4219ac3253c723c06c20,

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.",

year = "1999",

month = jan,

day = "20",

doi = "10.1006/jdeq.1998.3483",

language = "English",

volume = "151",

pages = "420--455",

number = "2",

}

TY - JOUR

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.

UR - http://www.scopus.com/inward/record.url?scp=0033585243&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033585243&partnerID=8YFLogxK

U2 - 10.1006/jdeq.1998.3483

DO - 10.1006/jdeq.1998.3483

M3 - Article

AN - SCOPUS:0033585243

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

Bibliographical note

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this