On the number of negative eigenvalues for a Schrödinger operator with magnetic field

Zhongwei Shen

doi:10.1007/BF02506420

On the number of negative eigenvalues for a Schrödinger operator with magnetic field

Zhongwei Shen

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Abstract

We consider the Schrödinger operator with magnetic field H = ∑ⁿ_j=1 (1/i ∂/∂xj - a_j)² + V in ℝⁿ. Under certain conditions on the magnetic field B = curl a, we generalize the Fefferman-Phong estimates (Bull. A. M. S. 9, 129-206 (1983)) on the number of negative eigenvalues for -Δ + V to the operator H. Upper and lower bounds are established. Our estimates incorporate the contribution from the magnetic field. The conditions on B in particular are satisfied if the magnetic potentials a_j(x) are polynomials.

Original language	English
Pages (from-to)	637-660
Number of pages	24
Journal	Communications in Mathematical Physics
Volume	182
Issue number	3
DOIs	https://doi.org/10.1007/BF02506420
State	Published - 1996

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/BF02506420

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title = "On the number of negative eigenvalues for a Schr{\"o}dinger operator with magnetic field",

abstract = "We consider the Schr{\"o}dinger operator with magnetic field H = ∑nj=1 (1/i ∂/∂xj - aj)2 + V in ℝn. Under certain conditions on the magnetic field B = curl a, we generalize the Fefferman-Phong estimates (Bull. A. M. S. 9, 129-206 (1983)) on the number of negative eigenvalues for -Δ + V to the operator H. Upper and lower bounds are established. Our estimates incorporate the contribution from the magnetic field. The conditions on B in particular are satisfied if the magnetic potentials aj(x) are polynomials.",

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year = "1996",

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T1 - On the number of negative eigenvalues for a Schrödinger operator with magnetic field

AU - Shen, Zhongwei

PY - 1996

Y1 - 1996

N2 - We consider the Schrödinger operator with magnetic field H = ∑nj=1 (1/i ∂/∂xj - aj)2 + V in ℝn. Under certain conditions on the magnetic field B = curl a, we generalize the Fefferman-Phong estimates (Bull. A. M. S. 9, 129-206 (1983)) on the number of negative eigenvalues for -Δ + V to the operator H. Upper and lower bounds are established. Our estimates incorporate the contribution from the magnetic field. The conditions on B in particular are satisfied if the magnetic potentials aj(x) are polynomials.

AB - We consider the Schrödinger operator with magnetic field H = ∑nj=1 (1/i ∂/∂xj - aj)2 + V in ℝn. Under certain conditions on the magnetic field B = curl a, we generalize the Fefferman-Phong estimates (Bull. A. M. S. 9, 129-206 (1983)) on the number of negative eigenvalues for -Δ + V to the operator H. Upper and lower bounds are established. Our estimates incorporate the contribution from the magnetic field. The conditions on B in particular are satisfied if the magnetic potentials aj(x) are polynomials.

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UR - https://www.scopus.com/pages/publications/0030569491#tab=citedBy

U2 - 10.1007/BF02506420

DO - 10.1007/BF02506420

M3 - Article

AN - SCOPUS:0030569491

SN - 0010-3616

VL - 182

SP - 637

EP - 660

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -

On the number of negative eigenvalues for a Schrödinger operator with magnetic field

Abstract

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