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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Journal of Differential Equations
( Vol. 149 NO.2 / (1998))
On Moments of Negative Eigenvalues for the Pauli Operator
Zhongwei Shen,
Pages. 292-327
Abstract This paper concerns the three-dimensional Pauli operator $$Bbb P =
(sigma cdot({f p}- {f A}(x)))^2 + V(x)$$ with a
nonhomogeneous megnetic field ${f B} = {
m curl} f A$. The
following Lieb-Thirring type inequality for the moment of negative
eigenvalues is established as
$$sum_{lambda_j<0}|lambda_j|
leqslant C_1 int_{Bbb R^3} |V(x)|^{5/2}_{-} dx + C_2int_{Bbb
R^3} [b_p(x)]^{3/2} |V(x)|_{-} dx,$$ where $q>3/2$ and $b_p(x)$ is
the $L^p$ average of $|f B|$ over certain cube centered at $x$
with a side scaling like $|{f B}|^{-1/2}$. We also show that, if
$f B$ has a constant direction,
$$sum_{lambda_j<0}|lambda_j|^gamma leqslant C_{1,gamma} int_{Bbb R^3}
|V(x)|^{gamma+3/2}_{-} dx + C_{2,gamma}int_{Bbb R^3}
b_p(x)|V(x)|^{gamma+1/2}_{-} dx,$$ where $gamma > 1/2$ and $p >
1$.
Contents 1. Introduction
2. Some preliminaries
3. A partition of unity
4. The proof of theorem 1.1
5. Fields with constant directions
Key words
Mathmatical Subject Classification