On Moments of Negative Eigenvalues for the Pauli Operator

✍ Scribed by Zhongwei Shen

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 471 KB
• Volume: 149
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1998.3473

✦ Synopsis

This paper concerns the three-dimensional Pauli operator P=(_ } (p&A(x))) 2 +V(x) with a nonhomogeneous magnetic field B=curl A. The following Lieb Thirring type inequality for the moment of negative eigenvalues is established as

where p>3Â2 and b p (x) is the L p average of |B| over certain cube centered at x with a side length scaling like |B| &1Â2 . We also show that, if B has a constant direction, :

where #>1Â2 and p>1.

1998 Academic Press acting on L 2 (R 3 , C 2 ). Here _=(_ 1 , _ 2 , _ 3 ) denotes the vector of Pauli matrices, p=&i {, and

) is a vector potential. We shall use B=curl A to denote the magnetic field article no.